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^t^-A r^'T^^?'-^- 



HARVARD COLLEGE 
LIBRARY 




FROM THE 

FARRAR FUND 



rA« fttigiMVf oj Iff*. WHuL Farrat tR 
memory of her husband^ John FoTrar, 
HoUi$ FnfBuoir rf Math e mnOioBt 
Attromomiif and Natural PkUoaophif, 
t907-18S6 




CUxjU.. |o. ns 



o 



Karl Friedrich Gauss 



General Investigations 



OF 



Curved Surfaces^^ 



OF 



1827 AND 1825 



translated with notes 

AND A 

bibliography 

BY 

JAMES CADDALL MOREHEAD, A.M.. M.S., and ADAM MILLER HILTEBEITEL, A.M. 

J. S. K. FELLOWS IN MATHEMATICS IN PRINCETON UNIYERSITY 



THE PRINCETON UNIVERSITY LIBRARY 

1902 



"VVkOts. *ioo^^xi.x 



' . . • 




^"jfiyV>.£Cr\^ J lAAs.^ 



Copyright, 1902, by 
The Princeton Uniyebsity Libraby 



C. 8. Robimon & Oo.^ Vnvoerrity Press 
PrinceUmy N. J, 



INTRODUCTION 

In 1827 Gauss presented to the Rpyal Society of Gottingen his important paper on 
the theory of surfaces, which seventy-three years afterward the eminent French 
geometer, who has done more than any one else to propagate these principles, charac- 
terizes as one of Gauss's chief titles to fame, and as still the most finished and use-' 
fill introduction to the study of infinitesimal geometry \ This memoir may be called: 
General Investigations of Curved Surfaces, or the Paper of 1827, to distinguish it 
fi'om the original draft written out in 1825, but not published until 1900. A list of 
the editions and translations of the Paper of 1827 follows. There are three editions 
in Latin, two translations into French, and two into German. The paper was origin- 
ally published in Latin under the title : 

la. Disquisitiones generales circa superficies curvas 
auctore Carolo Friderico Gauss 

Societati regise oblatae D. 8. Octob. 1827, 

and was printed in: Commentationes societatis regise scientiarum Gottingensis recen- 
tiores, Commentationes classis mathematicae. Tom. VI. (ad a. 1823-1827). Gottingse, 
1828, pages 99-146. This sixth volume is rare; so much so, indeed, that the British 
>.Museum Catalogue indicates that it is missing in that collection. With the signatures 
changed, and the paging changed to pages 1-50, la also appears with the title page 
added : 

lb. Disquisitiones generales circa superficies curvas 
auctore Carolo Friderico Gauss. 

Gotting». Typis Dieterichianis. 1828. 

II. In Monge's Application de Tanalyse h la g6om6trie, fifth edition, edited by 
Liouville, Paris, 1850, on pages 505-546, is a reprint, added by the Editor, in Latin 
under the title: Recherches sur la th^orie g^n^rale des surfaces courbes; Par M. 
C.-F. Gauss. 



' G. Darboux, Bulletin des Sciences Math. Ser. 2, vol. 24, page 278, 1900. 



\ 




iv INTRODUCTION 

Ilia. A third Latin edition of this paper stands in: Gauss, Werke, Herausge- 
geben von der Koniglichen Gesellschaft der Wissenschaften zu Gottingen, Vol. 4, Got- 
tingen, 1873, pages 217-258, without change of the title of the original paper (la). 

Ill J. The same, without change, in Vol. 4 of Gauss, Werke, Zweiter Abdruck, 
Gottingen, 1880. 

IV. A French translation was made from Liouville's edition, II, by Captain 
Tiburce Abadie, ancien 616 ve de Tficole Poly technique, and appears in Nouvelles 
Annales de Math^matique, Vol. 11, Paris, 1852, pages 195-262, under the title : 
Recherches g^n^rales sur les surfaces courbes ; Par M. Gauss. This latter also 
appears under its own title. 

Va. Another French translation is : Recherches G^n^rales sur les Surfaces 
Courbes. Par M. C.-F. Gauss, traduites en fran§ais, suivies de notes et d'^tudes 
sur divers points de la Th^orie des Surfaces et sur certaines classes de Courbes, par 
M. B. Roger, Paris, 1855. 

V*. The same. Deuxi^me Mtion. Grenoble (or Paris), 1870 (or 1871), 160 
pages. 

VI. A German translation is the first portion of the second part, namely, pages 
198-232, of: Otto Boklen, Analytische Geometric des Raumes, Zweite Auflage, Stutt- 
gart, 1884, under the title (on page 198) : Untersuchungen liber die allgemeine Theorie 
der krummen Flachen. Von C. F. Gauss. On the title page of the book the second 
part stands as : Disquisitiones generales circa superficies curvas von C. F. Gauss, ins 
Deutsche ubertragen mit Anwendungen und Zusatzen .... 

Vila. A second German translation is No. 5 of Ostwald's Klassiker der exacten 
Wissenschaften : AUgemeine Flachentheorie (Disquisitiones generales circa superficies 
curvas) von Carl Friedrich Gauss, (1827). Deutsch herausgegeben von A. Wangerin. 
Leipzig, 1889. 62 pages. 

VII J. The same. Zweite revidirte Auflage. Leipzig, 1900. 64 pages. 

The English translation of the Paper of 1827 here given is from a copy of the 
original paper, la; but in the preparation of the translation and the notes aU the 
other editions, except Va, were at hand, and were used. The excellent edition of 
Professor Wangerin, VII, has been used throughout most freely for the text and 
notes, even when special notice of this is not made. It has been the endeavor of 
the translators to retain as far as possible the notation, the form and punctuation of 
the formulae, and the general style of the original papers. Some changes have been 
made in order to conform to more recent notations, and the most important of these 
are mentioned in the notes. 



mTRODUCTION 

The second paper, the translation of which is here given, is the abstract (Anzeige) 
which Gauss presented in German to the Royal Society of Gottingen, and which was 
published in the Gottingische gelehrte Anzeigen. Stxick 177. Pages 1761-1768. 1827. 
November 5. It has been translated into English from pages 341-347 of the fourth 
volume of Gauss's Works. This abstract is in the nature of a note on the Paper of 
1827, and is printed before the notes on that paper. 

Recently the eighth volume of Gauss's Works has appeared. This contains on 
pages 408-442 the paper which Gauss wrote out, but did not publish, in 1825. This 
paper may be called the New General Investigations of Curved Surfaces, or the Paper 
of 1825, to distinguish' it from the Paper of 1827. The Paper of 1825 shows the 
manner in which many of the ideas were evolved, and while incomplete and in some 
cases inconsistent, nevertheless, when taken in connection with the Paper of 1827, 
shows the development of these ideas in the mind of Gauss. In both papers are 
found the method of the spherical representation, and, as types, the three important 
theorems : The measure of curvature is equal to the product of the reciprocals of the 
principal radii of curvature of the surface. The measure of curvature remains unchanged 
by a mere bending of the surface. The excess of the sum of the angles of a geodesic 
triangle is measured by the area of the corresponding triangle on the auxiliary sphere. 
But in the Paper of 1825 the first six sections, more than one-fifth of the whole paper, 
take up the consideration of theorems on curvature in a plane, as an introduction, 
before the ideas are used in space ; whereas the Paper of 1827 takes up these ideas 
for space only. Moreover, while Gauss introduces the geodesic polar coordinates in 
the Paper of 1825, in the Paper of 1827 he uses the general coordinates, jt?, y, thus 
introducing a new method, as well as employing the principles used by Monge and 
others. 

The publication of this translation has been made possible by the liberality of 
the Princeton Library Publishing Association and of the Alumni of the University 
who founded the Mathematical Seminary. 

H. D. Thompson. 



Mathjbmaticai. Ssmikabt, 

Pbikoktok Ukitsbsitt Libsabt, 

January 29, 1902. 



1, 






CONTENTS 



Grauss's Paper of 1827, General Investigations of Curved Surfaces . 



PAGE 



• • 



Gauss's Abstract of the Paper of 1827 45 

Notes on the Paper of 1827 51 

Gauss's Paper of 1825, New General Investigations of Curved Surfaces . 79 

Notes on the Paper of 1825 Ill 

Bibliography of the General Theory of Surfaces 115 



DISQUISITIONES GENERALES 



CIRCA 



SUPERFICIES CURVAS 



AUCTORE 



CAROLO FRIDERICO GAUSS 



SOCIETATI REGIAE OBLATAE D. 8. OCTOB. 1827 



COMMENTATIONES SOCIETATIS REGIAE SCIENTIARUM 
GOTTINGENSIS RECENTIORES. VOL. VI. GOTTINGAE MDCCCXXVIII 



GOTTINGAE 

TYPIS DIETERICHIANIS 
MDCCCXXVm 



GENERAL INVESTIGATIONS 



OF 



CURVED SURFACES 



BY 



KARL FRIEDRICH GAUSS 



PRESENTED TO THE ROTAL SOCIETT, OCTOBER 8, 1827 



1. 

Investigations, in which the directions of various straight lines in space are to be 
considered, attain a high degree of clearness and simplicity if we employ, as an auxil- 
iary, a sphere of unit radius described about an arbitrary centre, and suppose the 
different points of the sphere to represent the directions of straight lines parallel to 
the radii ending at these points. As the position of every point in space is deter- 
mined by three coordinates, that is to say, the distances of the point from three mutually 
perpendicular fixed planes, it is necessary to consider, first of all, the directions of the 
axes perpendicular to these planes. The points on the sphere, which represent these 
directions, we shall denote by (1), (2), (3). The distance of any one of these points 
from either of the other two will be a quadrant; and we shall suppose that the direc- 
tions of the axes are those in which the corresponding coordinates increase. 

2. 

It will be advantageous to bring together here some propositions which are fre- 
quently used in questions of this kind. 

I. The angle between two intersecting straight lines is measured by the arc 
between the points on the sphere which correspond to the directions of the lines. 

II. The orientation of any plane whatever can be represented by the great circle 
on the sphere, the plane of which is parallel to the given plane. 



4 KABL FRIEDBICH GAUSS 

m. The angle between two planes is equal to the spherical angle between the 
great circles representing them, and/ consequently , is also measured by the arc inter- 
cepted between the poles of these great circles. And, in like manner, the angle of inclina- 
tion of a straight line to a plane is measured by the arc drawn from the point which 
corresponds to the direction of the line, perpendicular to the great circle which repre- 
sents the orientation of the plane. 

lY. Letting :r, y, z ; ar', y, 0' denote the coordinates of two points, r the distance 
between them, and L the point on the sphere which represents the direction of the line 
drawn from the first point to the second, we shall have 

7fr=z a? + r cos (l)-C 
!^=y + r cos(2)Z 
0^= « + r cos (3)-C 

y. From this it follows at once that, generally, 

cos" {1)L + cos* (2)i; + COS* {%)L = 1 

and also, if L' denote any other point on the sphere, 

cos (l)ii . COS (1)X'+ COS (2)Z • cos (2)ii'+ cos (3)i . cos (3)i'= cos LL'. 

VI. Theorem. If Z, L'y L"y L" denote four paints an the sphere, and A the angle 
which the arcs LL'y L"L'" make at their paint of'intersectiany then toe shall have 

cos LL" . cos L'r"— cos LL"' . cos L'r'= sin LL' . sin L''L''' . cos A 

Demmsiration. Let A denote also the point of intersection itself, and set 

AL^ty AV^f, AL"^f\ AL"'=f'' 

Then we shall have 

cos LL" = cos ^ . cos <" + sin < sin f cos A 
cos L'L"'= cos a cos <"'+ sin H sin f" cos A 
cos LL'" = cos t cos f" + &iat sin f" cos A 
cos L'L" = cos f cos ^'^ + sin ^^ sin <" cos A 
and consequently, 

cos LL" . cos L'L'"— cos LL'". cos L'L" 

= cos A (cos t cos t" sin f sin f" + cos f cos f" sin t sin f 

— cos t cos f" sin f sin t" — cos f cos f sin t sin f") 
= cos A (cos t sin f — sin t cos f) (cos t" sin f" — sin f cos f") 
= cos il . sin (f—t) . sin {f"—t") 
= cos il . sin LL' . sin L"L"' 



GENERAL INVESTIGATIONS OF CUBVED SURFACES 6 

But as there are for each great circle two branches going out from the point Ay 
these two branches form at this point two angles whose sum is 180^. But our analysis 
shows that those branches are to be taken whose directions are in the sense from the 
point L to L\ and from the point L'^ to L^'^; and since great circles intersect in two 
points, it is clear that either of the two points can be chosen arbitrarily. Also, instead 
of the angle Ay we can take the arc between the poles of the great circles of which the 
arcs L L\ V L'" are parts. But it is evident that those poles are to be chosen which 
are similarly placed with respect to these arcs ; that is to say, when we go from L to L' 
and from L'^ to L"'y both of the two poles are to be on the right, or both on the left. 

VII. Let Ly Vy L" be the three points on the sphere and set, for brevity, 

cos (l)-C == Xy COS (2)Z = y, cos (3)-C = 2 
cos {1)L' = a/y cos {2)L' = /, cos {S)r = 0' 
cos (1)X''= x^'y cos {2)r'= y', cos (3)Z"= 0^' 

and also 

xy' a'' + z'y" g + 7f' y^ — xjf' ff —txf y ^' — txf' if z = A 

Let X denote the pole of the great circle of which LL' is a part, this pole being the one 
that is placed in the same position with respect to this arc as the point (1) is with 
respect to the arc (2)(3). Then we shall have, by the preceding theorem, 

y ^ — y z = cos (1)X . sin (2)(3) • sinZZ', 

or, because (2)(3) = 90^, 

y ff — ^ z = cos (1)X . sin LL\ 
and similarly, 

z 7f — ^ rr = COS (2)X . sinZX' 
xif — 9f y = cos (3)X . sin LV 

Multipl3ring these equations by 9f'y y , ^' respectively, and adding, we obtain, by means 
of the second of the theorems deduced in V, 

A =r cos X L'' . sin LL' 

Now there are three cases to be distinguished. Fir^ily when L" lies on the great circle 
of which the arc LL' is a part, we shall have \L''= 90®, and consequently, A = 0. 
If L" does not lie on that great circle, the second case will be when L'' is on the same 
side as X ; the third case when they are on opposite sides. In the last two cases the 
points Ly L'y L" will form a spherical triangle, and in the second case these points will lie 
in the same order as the points (1), (2), (3), and in the opposite order in the third case. 



6 KABL FEIEDRICH GAUSS 

» 

Denoting the angles of this triangle simply by Z, L\ L" and the perpendicular drawn on 
the sphere from the point L" to the side LL' by jt?, we shall have 

sinjt? = sin Z . sin LL''= sin L' . sin L' L"y 
and 

the upper sign being taken for the second case, the lower for the third. From this 
it follows that 

± A = sin X . sin LL' . sin LL'^ = sin Z' . sin LL'. sin L'L" 
= sin L" . sin LL" . sin L'L" 

Moreover, it is evident that the first case can be regarded as contained in the second or 
third, and it is easily seen that the expression ± A represents six times the volume of 
the pyramid formed by the points Z, L'^L" and the centre of the sphere. Whence, 
finally, it is clear that the expression ± i A expresses generally the volume of any 
pyramid contained between the origin of coordinates and the three points whose coor- 
dinates are 0, y, z ; a/, y, vf ; a/', y , e^'. 

3. 

A curved surface is said to possess continuous curvature at one of its points Ay if the 
directions of all the straight lines drawn from A to points of the surface at an infinitely 
small distance from A are deflected infinitely little from one and the same plane passing 
through A. This plane is said to touch the surface at the point A. K this condition is 
not satisfied for any point, the continuity of the curvature is here interrupted, as happens, 
for example, at the vertex of a cone. The following investigations will be restricted to 
such surfaces, or to such parts of surfaces, as have the continuity of their curvature 
nowhere interrupted. We shall only observe now that the methods used to determine 
the position of the tangent plane lose their meaning at singular points, in which the 
continuity of the curvature is interrupted, and must lead to indeterminate solutions. 

4. 

The orientation of the tangent plane is most conveniently studied by means of the 
direction of the straight line normal to the plane at the point Ay which is also called the 
normal to the curved surface at the point A. We shall represent the direction of this 
normal by the point L on the auxiliary sphere, and we shall set 

cos (1)Z =Xy COS {2)L = F, cos (3)Z =Z; 

and denote the coordinates of the point A by Xy y, z. Also let x + dxy y + rfy, z-^ dz 
be the coordinates of another point A' on the curved surface ; d^ its distance from Ay 



GENERAL INVESTIGATIONS OF CURVED SURFACES 7 

which is infinitely small ; and finally, let X be the point on the sphere representing the 
direction of the element A A\ Then we shall have 

dx = d8. cos (1)X, dy = d8. cos (2)X, dz = d8.co^ (3)X 

and, since X L must be equal to 90^, 

X cos (1)X + Fcos (2)X + Z cos (3)X = 

By combining these equations we obtain 

Xdx + Ydy+Zdz = Q. 

There are two general methods for defining the nature of a curved surface. The 
first uses the equation between the coordinates Xy y^ z^ which we may suppose reduced to 
the form TF = 0, where W will be a function of the indeterminates x^ y^ z. Let the com- 
plete difierential of the function W be 

d W= Pdx+ Qdy + Rdz 

and on the curved surface we shall have 

Pdx+ Qdy + Rdz = 
and consequently, 

P cos (1)X + Q cos (2)X + R cos (3)X = 

Since this equation, as well as the one we have established above, must be true for the 
directions of all elements ds on the curved surface, we easily see that X, JT, Z must be 
proportional to P, Q, R respectively, and consequently, since 

x^+jr + z^=i, 

we shall have either 



or 



l/(P»+C*+^j ~i/(P»+C*+^) ^"l/(i^+C*+^) 

The %ec(md method expresses the coordinates in the form of functions of two variar 
bles, jp, q. Suppose that differentiation of these functions gives 

dx z= a dp -■¥ a' dq 
dy = b dp + y dq 
dz = e dp + (/ dq 



8 KABL FRIEDBIOH GAUSS 

Substituting these values in the fonnula given above^ we obtain 

{aX+b Y+eZ)dp +{a' X+y Y+c" Z)dq = 

Since this equation must hold independently of the values of the differentials dp^ dq^ 
we evidently shall have 

From this we see that Xy Yy Z will be proportioned to the quantities 

hcf — eVy ea' — «(?', aV — ha* 
Hence, on setting, for brevity, 

1/ ( (J c' — c? JO* + (^«' — «0* + («*' — *«0") = -^ 

we shall have either 

^ hif — cV ca!—a^ ^_ <^'—i^' 
X- ^ — , Y ^ , Z ^ 

or 

cV — be' wf—ea! ^ ha'— ah' 

X- ^— , Y- ^— , Z- ^— 

With these two general methods is associated a thirds in which one of the coordinates, 
Zj say, is expressed in the form of a function of the other two, x, y. This method is 
evidently only a particular case either of the first method, or of the second. If we set 

dzzsztdx-^ udy 
we shall have either 

— t — u 1 



or 

t i u — 1 

^"'i/(l + <« + u«/ ^""i/a + ^^ + u")' ^"^ 1/ (1 + <* + u") 

The two solutions found in the preceding article evidently refer to opposite points of 
the sphere, or to opposite directions, as one would expect, since the normal may be drawn 
toward either of the two sides of the curved surface. If we wish to distinguish between 
the two regions bordering upon the surface, and call one the exterior region and the other 
the interior region, we can then assign to each of the two normals its appropriate solution 
by aid of the theorem derived in Art. 2 (VII), and at the same time establish a criterion 
for distinguishing the one region from the other. 



GENERAL INVESTIGATIONS OF CUBVED SUBFACES 9 

In the first method, such a criterion is to be drawn from the sign of the quantity W. 
Indeed, generally speaking, the curved surface divides those regions of space in which W 
keeps a positive value from those in which the value of W becomes negative. In fact, it 
is easily seen from this theorem that, if W takes a positive value toward the exterior 
region, and if the normal is supposed to be drawn outwardly, the first solution is to be 
taken. Moreover, it will be easy to decide in any case whether the same rule for the 
sign of TF is to hold throughout the entire surface, or whether for different parts there 
will be different rules. As long as the coefficients P, Q, B have finite values and do not 
all vanish at the same time, the law of continuity will prevent any change. 

K we follow the second method, we can imagine two systems of curved lines on the 
curved surface, one system for which p is variable, q constant ; the other for which q is 
variable, jp constant. The respective positions of these lines with reference to the exte- 
rior region will decide which of the two solutions must be taken. In fact, whenever 
the three lines, namely, the branch of the line of the former system going out from the 
point A BBp increases, the branch of the line of the latter system going out from the point 
Am q increases, and the normal drawn toward the exterior region, are similarly placed as 
the Xy jfj z axes respectively from the origin of abscissas {e, g., if, both for the former 
three lines and for the latter three, we can conceive the first directed to the left, the 
second to the right, and the third upward), the first solution is to be taken. But when- 
ever the relative position of the three lines is opposite to the relative position of the 
Xj y, z axes, the second solution will hold. 

In the third method, it is to be seen whether, when z receives a positive increment, x 
and y remaining constant, the point crosses toward the exterior or the interior region. 
In the former case, for the normal drawn outward, the first solution holds ; in the latter 
case, the second. 

6. 

Just as each definite point on the curved surface is made to correspond to a definite 
point on the sphere, by the direction of the normal to the curved surface which is trans- 
ferred to the surfiGtce of the sphere, so also any line whatever, or any figure whatever, on 
the latter will be represented by a corresponding line or figure on the former. In the 
comparison of two figures corresponding to one another in this way, one of which will be 
as the map of the other, two important points are to be considered, one when quantity 
alone is considered, the other when, disregarding quantitative relations, position alone 
is considered. 

The first of these important points will be the basis of some ideas which it seems 
judicious to introduce into the theory of curved surfaces. Thus, to each part of a curved 



10 KARL PRIEDEIOH GAUSS 

surface inclosed within definite limits we assign a total or integral curvature^ which is 
represented by the area of the figure on the sphere corresponding to it. From this 
integral curvature must be distinguished the somewhat more specific curvature which we 
shall call the measure of curvature. The latter refers to a paint of the surface^ and shall 
denote the quotient obtained when the integral curvature of the surface element about 
a point is divided by the area of the element itself; and hence it denotes the ratio of the 
infinitely small areas which correspond to one another on the curved surface and on the 
sphere. The use of these innovations will be abundantly justified, as we hope, by what 
we shall explain below. As for the terminology, we have thought it especially desirable 
that all ambiguity be avoided. For this reason we have not thought it advantageous to 
follow strictly the analogy of the terminology commonly adopted (though not approved by 
all) in the theory of plane curves, according to which the measure of curvature should be 
called simply curvature, but the total curvature, the amplitude. But why not be free in 
the choice of words, provided they are not meaningless and not liable to a misleading 
interpretation ? 

The position of a figure on the sphere can be either similar to the position of the 
corresponding figure on the curved surface, or opposite (inverse). The former is the case 
when two lines going out on the curved surface from the same point in different, but not 
opposite directions, are represented on the sphere by lines similarly placed, that is, when 
the map of the line to the right is also to the right ; the latter is the case when the con- 
trary holds. We shall distinguish these two cases by the positive or negative sign of the 
measure of curvature. But evidently this distinction can hold only when on each surface 
we choose a definite face on which we suppose the figure to lie. On the auxiliary sphere 
we shall use always the exterior face, that is, that turned away from the centre ; on the 
curved surface also there may be taken for the exterior face tiie one already considered, 
or rather that face from which the normal is supposed to be drawn. For, evidently, there 
is no change in regard to the similitude of the figures, if on the curved surface both the 
figure and the normal be transferred to the opposite side, so long as the image itself 
is represented on the same side of the sphere. 

The positive or negative sign, which we assign to the measure of curvature accord- 
ing to the position of the infinitely small figure, we extend also to the integral curvature 
of a finite figure on the curved surface. However, if we wish to discuss the general case, 
some explanations will be necessary, which we can only touch here briefly. So long 
as the figure on the curved surface is such that to distinct points on itself there corres- 
pond distinct points on the sphere, the definition needs no further explanation. But 
whenever this condition is not satisfied, it will be necessary to take into account twice 
or several times certain parts of the figure on the sphere. Whence for a similar, or 



GENERAL INVESTiaATIONS OF CUBVED SURFACES 11 

inverse position, may arise an accumulation of areas, or the areas may partially or 
wholly destroy each other. In such a case, the simplest way is to suppose the curved 
surface divided into parts, such that each part, considered separately, satisfies the above 
condition ; to assign to each of the parts its integral curvature, determining this magni- 
tude by the area of the corresponding figure on the sphere, and the sign by the posi- 
tion of this figure; and, finally, to assign to the total figure the integral curvature 
arising from the addition of the integral curvatures which correspond to the single parts. 
So, generally, the integral curvature of a figure is equal to /Ar^or, dcr denoting the 
element of area of the figure, and k the measure of curvature at any point. The prin- 
cipal points concerning the geometric representation of this integral reduce to the fol- 
lowing. To the perimeter of the figure on the curved surface (under the restriction 
of Art. 3) will correspond always a closed line on the sphere. If the latter nowhere 
intersect itself, it will divide the whole surface of the sphere into two parts, one of 
which will correspond to the figure on the curved surface; and its area (taken as 
positive or negative according as, with respect to its perimeter, its position is similar, 
or inverse, to the position of the figure on the curved surface) will represent the inte- 
gral curvature of the figure on the curved surface. But whenever this line intersects 
itself once or several times, it will give a complicated figure, to which, however, it is 
possible to assign a definite area as legitimately as in the case of a figure without 
nodes; and this area, properly interpreted, will give always an exact value for the 
integral curvature. However, we must reserve for another occasion the more extended 
exposition of the theory of these figures viewed from this very general standpoint. 

7. 

We shall now find a formula which will express the measure of curvature for 
any point of a curved surface. Let dcr denote the area of an element of this surface ; 
then Zdcr will be the area of the projection of this element on the plane of the coor- 
dinates Xy y ; and consequently, if c? S is the area of the corresponding element on the 
sphere, Zd^ will be the area of its projection on the same plane. The positive or 
negative sign of Z will, in fact, indicate that the position of the projection is similar or 
inverse to that of the projected element. Evidently these projections have the same 
ratio as to quantity and the same relation as to position as the elements themselves. 
Let us consider now a triangular element on the curved surface, and let us suppose 
that the coordinates of the three points which form its projection are 

^, y 

X + ef a?, y-^ dy 

a? + 8ar, y + By 



12 KARL PRIEDRICH GAUSS ' 

I 

The double area of this triangle will be expressed by the formula i 

dx.hy — dy .hz i 

and this will be in a positive or negative form according as the position of the side I 

from the first point to the third, with respect to the side from the first point to the 
second, is similar or opposite to the position of the y-axis of coordinates with respect 
to the :r^xis of coordinates. 

In like manner, if the coordinates of the three points which form the projection of 
the corresponding element on the sphere, from the centre of the sphere as origin, are 

Jf, Y 

X+dX, Y+dY 
X+BX, Y+BY 

the double area of this projection will be expressed by 

dX.BY—dY.BX 

and the sign of this expression is determined in the same manner as above. Where- 
fore the measure of curvature at this point of the curved surface will be 

j^^ dX.BY—dY.SX ^^vi^ 

dx .hy — dy.hz '^< V 

If now we suppose the nature of the curved surface to be defined according to the third i 

method considered in Art. 4, X and Y will be in the form of functions of the quanti- i 

ties Xy y. We shall have, therefore, , 

dY dY 

^„ dY^ 'oY^ 
ax ay " 

When these values have been substituted, the above expression becomes 

dX dY dX BY . .-^ 



& = 



dx dy dy dx 



GENERAL INVESTIGATIONS OF CURVED SURFACES 18 



Setting, as above, 



and also 



cz , dz 

=- 7» =77 = V 



or 



^. ^/ /{>'■•■ ■ 



or 



dt = Tdz + Udy, du==Udx+Vdff 
we have from the formulaB given above ^ / ..•/*^ -^" ^^ 

X=-<Z, r=-uZ, (1 +<«+«•) ^= 1 <£-- 
and hence 

(l + f+u')dZ+Z{tdt + udu) = 

dZ=—Z*{tdi + udu) 
dX==—Z*{l + t^)dt + Z*tudu 
^dr=+Z*tudt^Z'{l + f)du 

jy=z'(-{i+u')u+tuvy 

Snbstitating these values in the above expression, it becomes 

k = Z'{TV-^lP){l + f+ «^)= Z*{T r— IP) 
TV— IP 



and so 



(1 + f+ ««)• 



8. 



By a suitable choice of origin and axes of coordinates, we can easily make the 
values of the quantities t, u, U vanish for a definite point A. Indeed, the first two 



14 KABL PREEDRIOH GAUSS 

conditions will be fulfilled at once if the tangent plane at this point be taken for the 
ary-plane. K, further, the origin is placed at the point A itself, the expression for 
the coordinate e evidently takes the form 

where Q will be of higher degree than the second. Turning now the axes of x and y ^ 
through an angle M such that ^ 

tan2Jlf=-J^l- 

Jfo pro 

it is easily seen that there must result an equation of the form 

In this way the third condition is also satisfied. When this has been done, it is evi- 
dent that 

I. K the curved surface be cut by a plane passing through the normal itself and 

through the o^axis, a plane curve will be obtained, the radius of curvature of which 

1 
at the point A will be equal to -^y the positive or negative sign indicating that the 

curve is concave or convex toward that region toward which the coordinates are 
positive. 

II. In like manner -= will be the radius of curvature at the point A of the plane 

curve which is the intersection of the surface and the plane through the y-axis and 
the z^xia. 

III. Setting rr = r cos <^, y r= r sin <^, the equation becomes 

0= J (^cos»<^ + rsin»<^)r«+ n 

from which we see that if the section is made by a plane through the normal at A 
and making an angle <^ with the a?-axis, we shall have a plane curve whose radius of 
curvature at the point A will be 

1 

IV. Therefore, whenever we have T=Vy the radii of curvature in aU the normal ( 
planes will be equal. But if T and V are not equal, it is evident that, since for any \ 
value whatever of the angle <^, T cos* <^ + F sin* <^ falls between T and F, the radii of 
curvature in the principal sections considered in I. and II. refer to the extreme curva- 
tures; that is to say, the one to the maximum curvature, the other to the minimum, 



GENERAL INVESTIGATIONS OF CURVED SURFACES 16 

if T and V have the same sign. On the other hand^ one has the greatest convex 
curvature, the other the greatest concave curvature, if T and V have opposite signs. 
These conclusions contain ahnost all that the illustrious Euler was the first to prove 
on the curvature of curved sur&ces. 

V. The measure of curvature at the point A on the curved surface takes the 
very simple form 

k = TV, 
whence we have the 

Theorem. The measure of curvature at any paint whatever of the surface is equal to a 
fraction whose numerator is unity y and whose denominator is the product of the two extreme 
rada of curvature of the sections by normal planes. 

At the same time it is clear that the measure of curvature is positive for con- 
cavo-concave or convexo-convex surfaces (which distinction is not essential), but nega- 
tive for concavo-convex surfaces. K the surface consists of parts of each kind, then 
on the lines separating the two kinds the measure of curvature w^i^ to vanish. Later 
we shall make a detailed study of the nature of curved surfaces for which the meas- 
ure of curvature everywhere vanishes. 

9. 

The general formula for the measure of curvature given at the end of Art. 7 is 
the most simple of all, since it involves only five elements. We shall arrive at a 
more complicated formula, indeed, one involving nine elements, if we wish to use the 
first method of representing a curved surface. Keeping the notation of Art. 4, let us 
set also 

?^_p/ ^-nf ^-1?/ 



dy .dz ' dx.dz ^ * dx.dy 

SO that 

dP^P' dx + R"dy + Q^'dz 
dQ = R"dz-\-Qf dy +P"dz 
dR -=q' dx-\- P"dy +i?' dz 

P 

Now since ^~"~»> ^e ^'^ through differentiation 

S?dt = -RdP + PdR^{P Qf'—RP') dx + (PP"^RR") dy + {PRf—R Q") dz 



16 EARL FBIEDRICH GATJSS 

or, eliminating dz by means of the equation 

Fdx+Qdy+Rde = Oy 
jp dt={—IP P'+2PR Q"—!^ R) dx + {PRP"-¥ QR Q^'—P QRf—E?R') dy. 

In like manner we obtain 

I^du^{pRP"-\- qRq'—PQB!—BR!')dx-\-{--Bqf-\-l QRP"—(^R)dp 

From this we conclude that 

i?2'=-iPJP'+ 2 PRQ"^P*Rf 
R^U=PRP"+ QRQ"-PQR'-R^R" 
B?V=-R?(^-^1QRP"-(^Rf 

Substituting these values in the formula of Art. 7, we obtain for the measure of curv- 
ature k the following symmetric expression : 

(^•4- q^-^Ofk^P^iO^R-P'") + <^{P'R-Q^'^) +R{P'Q'-R"') 
+ 2 QR{Q"R'-P'P") + 2 PR{P"R"-Q'Q") + 2PQ {P"Q"-RfR") 

10. 

We obtain a still more complicated formula, indeed, one involving fifteen elements, 
if we follow the second general method of defining the nature of a curved surface. It 
is, however, very important that we develop this formula also. Retaining the nota- 
tions of Art. 4, let us put also 

3*3? 3^ a? . 9** 

y — o g — at y — /y/ 

Zf P* dp.dq P^ 3/ P 

df ^' dp.dq ^' 3^ 
and let us put, for brevily. 



First we see that 



or 



Adx+Bdff+ Cde = Oy 



f 



I 

4 
■ 



f 



4 
i 



GENERAL INVESTIGATIONS OF CURVED SURFACES 17 

Thus, inasmuch as s may be regarded as a function of Xy y^ we have 

if — ^ — _A 



1 dy~^~ O 

Then from the formulse 

4 dx = adp + a^ dq, dy = bdp + V dq, 

c we have 

Cdp = V dx — a' dy 

Cdq = —b dx + a dy 
Thence we obtain for the total differentials of ty u 






4 \ dp dpi \ dq dq 

1 If now we substitute in these fonnulse 

1 dA 

m dB 

I -g^ = a'y+co'-ay'-c'o 

-g- = a Y + c o"- a y"- </ a' 

dC 

y- = J'o +a/8' -Ao' -a'/8 

i ^ 

I and if we note that the values of the di£ferentials dtydu thus obtained must be equal, 
independently of the differentials dx, dy,iQ the quantities Tdx + Udy, Udx+Vdy 
I respectiyely, we shall find, after some sufficiently obvious transformations, 

' C^T=aAb"+fiBb'*+yCb'* 

-2a'Abb'-2fi^Bbb'-2'/Cbb' 
+ a"Ab'+fi^'B^+-/'Cb* 

I 

I 

I 

1 

i 

■1 



1 

1 



18 KASL FRIEDRICH GAITSS 

€fU=-aAafy-fiBa'b'-yCa'y 

+ a'A{ab'+ baf)+firB{ab'+ ba') + y C{ab'+ ba') 
— a"Aab — p'Bab-'/'Cab 

-la! Aaa'-1^ Baa'-iy Caa' 
+ a" A a«+ j8" B o«+ y" Ca* 

Hence, if we put, for the sake of brevity, 

ilo +^/8 +Cy =i> (1) 

il o' + 5 /S' + Cy = ly (2) 

Aa"+5/8"+Cy"=i>" (3) 

we shall have 

C^T-^DV^—1jybV-\-jy'l? 

(P U=-Da'b'+D' {ab'+ ba') -D" ab 

C^V=Da'*-'2,iyaa'+D"i^ 

From this we find, after the reckoning has been carried out, 

(n ^rp pr_ j^) = {DD"-D") {ab'- ba'f = {pD"-D") C^ 

and therefore the formula for the measure of curvature 

11. 

By means of the formula just found we are going to establish another, which may 
be counted among the most productive theorems in the theory of curved surfaces. 
Let us introduce the following notation : 

a'«+ *'*+<?"= 0! 

aa+J/8+<?y=«» (4) 

a o'+i j8'+c /=»»' (5) 

a a"+i i8"+cy"=»»" (6) 

a'a +i'/8 +c'y =» (7) 

a'a'+ft'j8'+cV=n' (8) 

«'a"+i'j8"+c'y"=»" (9) 



GENERAL INVESTIGATIONS OF CUBVED SURFACES 19 

Let us eliminate from the equations 1, 4, 7 the quantities /3, y, which is done by 
multiplying them by bcf—eV, b' C—c^B, cB — lC respectively and adding. In this 
way we obtain 

iA{p</-eV) + a{VC-</B) + a!{eB-bC)'ia 
= D{be'-eb') + m{b'0-e'B)+n{eB-bC) 

an equation which is easily transformed into 

AD = a^ + a{nF—m€f)+a'{mF—nI!) 

Likewise the elimination of a, y or <t, /3 from the same equations gives 

BD = fiA + b(nF—mG) + y{mF—nJE) 
Ci> = yA+ c lnF—m€f) + c' {mF—nF) 

Multiplying these three equations by a", ^', y" respectively and adding, we obtain 

Diy'={aa"+fifi!'+y'/')A+m"{nF-mCri+n"{mF-nF) . . . (10) 

If we treat the equations 2, 6, 8 in the same way, we obtain 

AD'= o' A + a {n'F-m' G) + a' {m'F- n'E) 
Biy-= ^L-Vb\n'F-m'CP)-\-V {m'F -n'E) 
Ciy = y'A+ c {n'F -m'G) + c' (m'F- n'F) 

and after these equations are multiplied by a', P, y respectively, addition gives 

i>"= (a'«+ j8"+ y'«)A + m' {n'F-m' G) + n'{m'F-n'E) 

A combination of this equation with equation (10) gives 

2>jr>"-i)'«= (aa"+ /8i8"+ yy"-a'*-/8'«-y'«)A 

+ E{n'^—nn") +F{nm"- 2 m'n'+ mn") + G{m'*- mm") 

It is clear that we have 

dE BE ^ . dF dF dG dG 

^ = 2«,, ^ = 2< ^-m'+n, j^=m"+n', ^ = 2«', ■^ = 2n", 

or 

^dU , ^31! „ dF dO 

dp cq dq dp 

dF , dE dG dG 

dp dq dp dq 

Moreover^ it is easily shown that we shall have 

^_ ^E ^F ^G 

df ^ dp.dq *■ 3/ 



20 KARL FMEDRICH GAUSS 

If we substitute these dijSerent expressions in the formula for the measure of curva- 
ture derived at the end of the preceding article, we obtain the following formula, which 
involves only the quantities JS^ Fy G and their differential quotients of the first and 
second orders : 

. „idE dG dE dG ^dE dF ^ ,dE dF ^dF dG\ 

\dp dq dq dp dq dq dp dq dp dp 9 

(dE dG dE 3F idE\^_ iS'E ^F ^G\ 

12. 

Since we always have 

da^+df+dz' = Edp^+2Fdp.dq+Gdf, 

it is clear that 

i/{Edp''+2Fdp.dq+Gdf) 

is the general expression for the linear element on the curved surface. The analysis 
developed in the preceding article thus shows us that for finding the measure of cur- 
vature there is no need of finite formulae, which express the coordinates w^ ff, z ss 
functions of the indeterminates p^ q ; but that the general expression for the magnitude 
of any linear element is sufficient. Let us proceed to some applications of this very 
important theorem. 

Suppose that our surface can be developed upon another surface, curved or plane, 
so that to each point of the former surface, determined by the coordinates x^ y, Zy will 
correspond a definite point of the latter surface, whose coordinates are ^r', y, e^. Evi- 
dently ixfy y, ef can also be regarded as functions of the indeterminates j9, ^, and there- 
fore for the element \/{diiif^+ dy'^+ de^^) we shall have an expression of the form 

i/(J^rf/+ 2 Fdp . dq + &df) 

where E'y F'y G' also denote functions of j9, q. But from the very notion of the devd- 
apment of one surface upon another it is clear that the elements corresponding to one 
another on the two surfaces are necessarily equal. Therefore we shall have identically 

E^E^y F=Fy G=G'. 

Thus the formula of the preceding article leads of itself to the remarkable 

Theobem. If a curved mrfaee is developed upon any other mrface whatevery the 
measure of curvature in each paint remains unchanged. 



GENERAL INVESTIGATIONS OF CURVED SURFACES 21 

Also it is evident that any jinite 'part whatever of the curved mr/aee mU retain the 
same integral curvature after development upon another mrface. 

Surfaces developable upon a plane constitute the particular case to which geom- 
eters have heretofore restricted their attention. Our theory shows at once that the 
measure of curvature at every point of such surfaces is equal to zero. Consequently, 
if the nature of these surfaces is defined according to the third method, we shall have 
at every point 

a criterion which, though indeed known a short time ago, has not, at least to our 
knowledge, commonly been demonstrated with as much rigor as is desirable. 

13. 

What we have explained in the preceding article is connected with a particular 
method of studying surfaces, a very worthy method which may be , thoroughly devel- 
oped by geometers. When a surface is regarded, not as the boundary of a solid, but 
as a flexible, though not extensible solid, one dimension of which is supposed to 
vanish, then the properties of the surface depend in part upon the form to which we 
can suppose it reduced, and in part are absolute and remain invariable, whatever may 
be the form into which the surface is bent. To these latter properties, the study of 
which opens to geometry a new and fertile field, belong the measure of curvature and 
the integral curvature, in the sense which we have given to these expressions. To 
these belong also the theory of shortest lines, and a great part of what we reserve to 
be treated later. From this point of view, a plane surface and a surface developable 
on a plane, e. ^., cylindrical surfaces, conical surfaces, etc., are to be regarded as essen- 
tially identical; and the generic method of defining in a general manner the nature of 
the surfaces thus considered is always based upon the formula 

i/(^rf/+ 2 Fdp . dq + Gdf), 

which connects the linear element with the two indeterminates p^ q. But before fol- 
lowing this study further, we must introduce the principles of the theory of shortest 
lines on a given curved surface. 

14. 

The nature of a curved line in space is generally given in such a way that the 
coordinates a;, ^, e corresponding to the dijSerent points of it are given in the form of 
functions of a single variable, which we shall call to. The length of such a line from 



22 KARL PRIEDRICH GAUSS 

an arbitrary initial point to the point whose coordinates are z^ y^ z^ is expressed by 
the integral 

If we suppose that the position of the line undergoes an infinitely small variation, so 
that the coordinates of the different points receive the variations %Xy Sffy Sz^ the varia- 
tion of the whole length becomes 

dx.dSx+dy .dSy + dz.dSz 



f 



Vida^+di/'+d^) 
which expression we can change into the form 

dx .hz+dtf ,Zy + dz.hz 

~Jv^'^V{d:^+df+d;?) ^^^'^V{di^+df+d^) ^^^ '^ V{di^+ df+ dzy 

We know that, in case the line is to be the shortest between its end points, all that 
stands under the integral sign must vanish. Since the line must lie on the given 
surface, whose nature is defined by the equation 

Pdx + Qdy+Rdz = % 
the variations hx, 8^, hz also must satisfy the equation 

Phx + Qhy+Rhz = % 
and from this it follows at once, according to well-known rules, that the differentials 

, dx dy dz 

^V{dt^+df+d;?y ^V{d:i^+df+d^y ^ V{d^+ df+ d^) 

must be proportional to the quantities P, Q, R respectively. Let dr be the element 
of the curved line; X the point on the sphere representing the direction of this ele- 
ment ; L the point on the sphere representing the direction of the normal to the curved 
surface ; finally, let f, 7;, J be the coordinates of the point X, and X, F, Z be those of 
the point L with reference to the centre of the sphere. We shall then have 

dx = ^dr, dy = ridry dz = ^dr 

from which we see that the above differentials become J^, drf, dl^. And since the 

quantities P, Q, R are proportional to X, F, Zy the character of shortest lines is 

expressed by the equations 

d$^_dri _ rfi 

X" Y ^ Z 



GENERAL INVESTIQATIONS OP CURVED SURFACES 28 

Moreover, it is easily seen that 

is equal to the small arc on the sphere which measures the angle between the direc- 
tions of the tangents at the beginning and at the end of the element dr^ and is thus 

dr 
equal to — > if p denotes the radius of curvature of the shortest line at this point. 

Thus we shall have 

pd^=Xdr^ pdri = Ydry pd^=Zdr 

15. 

Suppose that an infinite number of shortest lines go out from a given point A 
on the curved surface, and suppose that we distinguish these lines from one another 
by the angle that the first element of each of them makes with the first element of 
one of them which we take for the first. Let ^ be that angle, or, more generally, a 
function of that angle, and r the length of such a shortest line from the point A to 
the point whose coordinates are Xy y^ z. Since to definite values of the variables r, ^ 
there correspond definite points of the surface, the coordinates x^ y, z can be regarded 
as functions of r, ^. We shall retain for the notation il, Z, f, ly, J, X, Y^ Z the same 
meaning as in the preceding article, this notation referring to any point whatever on 
any one of the shortest lines. 

All the shortest lines that are of the same length r will end on another line 
whose length, measured from an arbitrary initial point, we shall denote by v. Thus v 
can be regarded as a function of the indeterminates r, ^, and if \' denotes the point 
on the sphere corresponding to the direction of the element dvy and also ^', 17,' ([' 
denote the coordinates of this point with reference to the centre of the sphere, we 
shall have 



d^'^^'d^' d4>^'^'d4/ a^""^'a^ 



From these equations and from the equations 



we have 



9£_^ 3y__ ?£.— r 
ar""^' dr"'^' dft'^^ 



dx dx dy 3i di 3£_,>>;. ^ ,. ,.,..3!i_^^ vw.3»: 



24 KARL FRIEDRICH GAUSS 

Let S denote the first member of this equation, which will also be a function of r, <^. 
Differentiation of S with respect to r gives 



d^ dx ^d-q dp ^ djdz ^ . 3(f+V+n 
dr dd> dv dip dr d<p dip 



But 



and therefore its differential is equal to zero ; and by the preceding article we have, 
if p denotes the radius of curvature of the line r, 

dr"^ p^ dr p^ dr~ Q 
Thus we have 

^=l.(Xf + rv+^iO •!? = -• cos zv-|5 = o 

dr p dip p dip 

since X' evidently lies on the great circle whose pole is L. From this we see that 

8 is independent of r, and is, therefore, a function of ^ alone. But for r = we evi- 

dv 
dently have r = 0, consequently tt = 0, and ^8'= independently of if>. Thus, in general, 

we have necessarily 8=0^ and so cos XX'=0, i. e.y XX'= 90^. From this follows the 
Theorem. If on a curved mrface an infinite number of shortest lines of equal length 
he drawn from the same initial pointy the lines joining their extremities wiU be normal to 
each of the lines. 

We have thought it worth while to deduce this theorem from the fundamental 
property of shortest lines ; but the truth of the theorem can be made apparent with- 
out any calculation by means of the following reasoning. Let AB, AB' be two 
shortest lines of the same length including at A an infinitely small angle, and let us 
suppose that one of the angles made by the element BB' with the lines BA^ B' A 
differs from a right angle by a finite quantity. Then, by the law of continuity, one 
will be greater and the other less than a right angle. Suppose the angle at B is 
equal to 90^ — oi, and take on the line AB b, point (7, such that 

BC=BB' . cosec oi. 

Then, since the infinitely small triangle BB' Cm&y be regarded as plane, we shall have 

CB'=^BC. coa tti. 



GENERAL INVESTIGATIONS OP CURVED SXJRPA0E8 26 

and consequently 

AC+CB'=AC+BC .co^ta=AB-BC .{l-QO^to)=AB'-BC .{l-<^osta), 

i. e.y the path from A to B' through the point is shorter than the shortest line, 
Q. K A. 

16. 

With the theorem of the preceding article we associate another, which we state 
as follows : If on a curved surface we imoffine any line whatever^ from the different points 
of which are drawn at right angles and toward the same side an infinite number of shortest 
lines of the same lengthy the curve which joins their other extremities will cut each of the 
lines at right angles. For the demonstration of this theorem no change need be made 
in the preceding analysis, except that <^ must denote the length of the given curve 
measured from an arbitrary point; or rather, a function of this length. Thus all of 
the reasoning will hold here also, with this modification, that S=0 for r = is 
now implied in the hypothesis itself. Moreover, this theorem is more general than 
the preceding one, for we can regard it as including the first one if we take for the 
given line the infinitely small circle described about the centre A. Finally, we may 
say that here also geometric considerations may take the place of the analysis, which, 
however, we shall not take the time to consider here, since they are snfficientiy 
obvious. 

17. 

We return to the formula 

i/{II df+ 2 Fdp .dq + G df\ 

which expresses generally the magnitude of a linear element on the curved surface, 
and investigate, first of all, the geometric meaning of the coefficients E^ JP, G. We 
have already said in Art. 6 that two systems of lines may be supposed to lie on the 
curved surface, p being variable, q constant along each of the lines of the one system ; 
and q variable, p constant along each of the lines of the other system. Any point 
whatever on the surface can be regarded as the intersection of a line of the first 
system with a line of the second; and then the element of the first line adjacent to 
this point and corresponding to a variation dp will be equal to VE . dp, and the 
element of the second line corresponding to the variation dq will be equal io V G . dq. 
Finally, denoting by a» the angle between these elements, it is easily seen that we 
shall have 



26 KARL FREEDRICH GAUSS 

Furthermore, the area of the surface element in the form of a parallelogram between 
the two lines of the first system, to which correspond g, 9 + dq^ and the two lines of 
the second system, to which correspond p^ p + dpy will be 

V{Ea-F^)dp.dq. 

Any line whatever on the curved surface belonging to neither of the two sys- 
tems is determined when p and q are supposed to be functions of a new variable, or 
one of them is supposed to be a function of the other. Let s be the length of such 
a curve, measured from an arbitrary initial point, and in either direction chosen as 
positive. Let 6 denote the angle which the element 

d8 = V{Edf+2 Fdp .dq+ Gdf) 

makes with the line of the first system drawn through the initial point of the ele- 
ment, and, in order that no ambiguity may arise, let us suppose that this angle is 
measured from that branch of the first line on which the values of p increase, and is 
taken as positive toward that side toward which the values of q increase. These con- 
ventions being made, it is easily seen that 

cos g . rfg = i/^ . rf« + 1/ Q^ , cos ft> . rfg == ^ ^^ t/ ^^ 

sin ».., = •«. sin „.., = lli^^^^T£. 

18. 

We shall now investigate the condition that this line he a shortest line. Since 
its length « is expressed by the integral 



8 =/ \/{Edf-\- 2 Fdp .dg+G df) 



the condition for a minimum requires that the variation of this integral arising from 
an infinitely small change in the position become equal to zero. The calculation, for 
our purpose, is more simply made in this case, if we regard j» as a function of q. 
When this is done, if the variation is denoted by the characteristic S, we have 



Zs 



/\^.df^^^.dp.dq + ^>dq\Zp + {2Edp-{-^Fdq)dZp 
2rfi 



Edp+Fdq 



ds '^^ + 




GENERAL INVESTIGATIONS OF CURVED SURFACES 27 

'^\ 2Ts '^ Ts / 

and we know that what is included under the integral sign must vanish independently 
of Sp. Thus we have 

dp ^ ^ dp ^ ^^ dp ^ d» 

= 2d8.d. VE . cos 6 
ds . dE . cos 6 



VE 



— 2d8.de. VE . sin 



=.iE^P±^M^-i/{£G-F').dp.dd 
JE 

This gives the following conditional equation for a shortest line : 



dF ,1 dG_ , 

dp' ^ 2' dp' ^ 



which can also be written 



From this equation, by means of the equation 

E dp . F 



oot0=^^r^,^-—^.y^ + 



\/{EG-F^) dq ' V{EG-F^) 

it is also possible to eliminate the angle 0y and to derive a differential equation of 
the second order between p and q^ which, however, would become more complicated 
and less useful for applications than the preceding. 

19. 

The general formulae, which we have derived in Arts. 11, 18 for the measure of 
curvature and the variation in the direction of a shortest line, become much simpler 
if the quantities j9, q are so chosen that the lines of the first system cut everywhere 



28 KARL FRIEDRIOH GAUSS 



orthogonally the lines of the second system; ». e., in such a way that we have gen- 
erally a» = 90^, or F=0. Then the formula for the measure of* curvature becomes 

A Mrtaf ET d-E dG ldG\^ dE dG idE\^ c^r:,r^i^E 9G\ 

dq dq \dpf dp dp \dqf \dq^ dp^f 

and for the variation of the angle 

Among the various cases in which we have this condition of orthogonality, the 
most important is that in which all the lines of one of the two systems, e. g.y the 
first, are shortest lines. Here for a constant value of q the angle 9 becomes equal to 
zero, and therefore the equation for the variation of 6 just given shows that we must 

l^ave — = 0, or that the coefficient E must be independent ot/ffj i. e., E must be 

either a constant or a function of p alone. It wUl be simplest to take for p 
the length of each line of the first system, which length, when all the lines of the 
first system meet in a point, is to be measured from this point, or, if there is no 
common intersection, from any line whatever of the second system. Having made 
these conventions, it is evident that p and q denote now the same quantities that 
were expressed in Arts. 15, 16 by r and <^, and that E=l. Thus the two preced- 
ing formulae become : 

AG'k=:(^f-2G^^ 



or, setting i/G = my 



dpf dj^ 

VG.de = -\.^.dq 

m dpr dp 



Generally speaking, m will be a function of py q, and mdq the expression for the ele- 
ment of any line whatever of the second system. But in the particular case where 
all the lines p go out from the same point, evidently we must have m = for j9 = 0. 
Furthermore, in the case under discussion we wUl take for q the angle itself which 
the first element of any line whatever of the first system makes with the element of 
any one of the lines chosen arbitrarily. Then, since for an infinitely small value of 
p the element of a line of the second system (which can be regarded as a circle 
described with radius p) is equal to pdq^ we shall have for an infinitely small value 

dm 
of pym=py and consequently, for j!? = 0, m = at the same time, and r— =!• 



GENERAL INVESTiaATIONS OP CURVED SURFACES 29 

20, 

We pause to investigate the case in which we suppose that p denotes in a gen- 
eral manner the length of the shortest line drawn from a fixed point A to any other 
point whatever of the surface, and q the angle that the first element of this line 
makes with the first element of another given shortest line going out from A. Let 
^ be a definite point in the latter line, for which ^ = 0, and C another definite point 
of the surface, at which we denote the value of q simply by A. Let us suppose the 
points By C joined by a shortest line, the parts of which, measured from B, we denote 
in a general way, as in Art. 18, hj s ; and, as in the same article, let us denote by 9 
the angle which any element ds makes with the element dp; finally, let us denote 
by d^yff the values of the angle 6 at the points -B, C. We have thus on the curved 
surface a triangle formed by shortest lines. The angles of this triangle at B and C 
we shall denote simply by the same letters, and B will be equal to 180*^ — ^, C io ff 
itself. But, since it is easily seen from our analysis that all the angles are supposed 
to be expressed, not in degrees, but by numbers, in such a way that the angle 57^17' 
ib"j to which corresponds an arc equal to the radius, is taken for the unit, we must set 

where 2ir denotes the circumference of the sphere. Let us now examine the integral 
curvature of this triangle, which is equal to 

^hdcTy 

da denoting a surface element of the triangle. Wherefore, since this element is ex- 
pressed by mdp . dq^ we must extend the integral 

ffmdp.dq 

over the whole surface of the triangle. Let us begin by integration with respect to 
Py which, because 

^^ 1 d^m 

gives 

dg. (comt-^), 

for the integral curvature of the area lying between the lines of the first system, to 
which correspond the values qy q + dq of the second indeterminate. Since this inte- 



80 KAHL PKCEDRICH GAUSS 

gral curvature must vanish for p ~0y the constant introduced by integration must be 
equal to the value of ^ for j9 = 0, t. e., equal to unity. Thus we have 

where for -^ must be taken the value corresponding to the end of this area on the 
line CB. But on this line we have, by the preceding article, 

dq 

whence our expression is changed into dq + dO, Now by a second integration, taken 
from y = to j' = il, we obtain for the integral curvature 

or 

A+B^C-ir. 

The integral curvature is equal to the area of that part of the sphere which cor- 
responds to the triangle, taken with the positive or negative sign according as the 
curved surface on which the triangle lies is concavo-concave or concavo-convex. For 
unit area will be taken the square whose side is equal to unity (the radius of the 
sphere), and then the whole surface of the sphere becomes equal to 4ir. Thus the 
part of the surface of the sphere corresponding to the triangle is to the whole surface 
of the sphere as ± {A+B + C — ii) is to 4ir. This theorem, which, if we mistake 
not, ought to be counted among the most elegant in the theory of curved surfaces, 
may also be stated as follows : 

The excess over 180^ of the sum of the angles of a triangle formed hy shortest lines 
on a concavo^oncave curved surface, or the deficit from 180^ of the sum of the angles of 
a triangle formed by shortest lines on a concavo^onvex curved surface, is measured hy the 
area of the part of the sphere which corresponds, through the directions of the normals, to 
that triangle, if the whole surface of the sphere is set equal to 720 degrees. 

More generally, in any polygon whatever of n sides, each formed by a shortest 
line, the excess of the sum of the angles over (2 n — 4) right angles, or the deficit from 
(2» — 4) right angles (according to the nature of the curved surface), is equal to the 
area of the corresponding polygon on the sphere, if the whole surface of the sphere is 
set equal to 720 degrees. This follows at once from the preceding theorem by divid- 
ing the polygon into triangles. 



GENERAL INVESTIGATIONS OP CURVED SURFACES 81 

21. 

Let us again give to the symbols p^ q^ Ej Fy Gj on the general meanings which 
were given to them above, and let us further suppose that the nature of the curved 
surface is defined in a similar way by two other variables, p'j q\ in which case the 
general linear element is expressed by 

l/(J^ dp'^+ 2 F dp'. dq'+ a' dq'^) 

Thus to any point whatever Ijring on the surface and defined by definite values of 
the variables p^ q will correspond definite values of the variables p\ /, which will 
therefore be functions of j9, q. Let us suppose we obtain by diflPerentiating them 

dp'^adp+fidq 
dq'=ydp + hdq 

We shall now investigate the geometric meaning of the coefficients fl^ /8, 7, 8. 

Now four systems of lines may thus be supposed to lie upon the curved surface, 
for which p^ y, jt?', / respectively are constants. If through the definite point to 
which correspond the values p, y, p\ ^ of the variables we suppose the four lines 
belonging to these different systems to be drawn, the elements of these lines, corres- 
ponding to the positive increments dp^ dq^ dp% dq\ will be 

VE.dp, VG. dq, VE'. dp', V G'. dq'. 

The angles which the directions of these elements make with an arbitrary fixed direc- 
tion we shall denote by JIf, N, JIT, i\r, measuring them in the sense in which the 
second is placed with respect to the first, so that ^m{N—M) is positive. Let us 
suppose (which is permissible) that the fourth is placed in the same sense with respect 
to^ the third, so that sin (iV— JIT) also is positive. Having made these conventions, 
if we consider another point at an infinitely small distance from the first point, and 
to which correspond the values jt> + i?p, q + dq, p'+ dp', q'+dq' of the variables, we 
see without much difficulty that we shall have generally, a. e., independently of the 
values of the increments dp, dq, dp', dq', 

VE.dp. sin JIf + VG.dq. smN= VE'. dp'. sinJIf + VG'. dq'. siniV 

since each of these expressions is merely the distance of the new point from the line 
from which the angles of the directions begin. But we have, by the notation intro- 
duced above, 

In like manner we set 

If-M'-^J. 



32 



KARL PRIEDRIOH GAUSS 



and also 

Then the equation just found can be thrown into the following form : 

VE.dp. sin {M—f,^ + ^) + \/G.dq. sin {M+ ^) 

= i/JET. df. sinilf + VG'. dq\ sin (i!f + a>') 
or 

VE.dp. sin (iV'—oi— 6i'+ ^-\-V G .dq .^m (iV— 6i'+ ^) 

= VM.dp'. sin {ISr—a/) + i/(y'. rf^. sin IT 

And since the equation evidently must be independent of the initial direction, this 
direction can be chosen arbitrarily. Then, setting in the second formula iV=0, or in 
the first -3/^=0, we obtain the following equations : 

\/I?. sin a>'. dp'=VE . sin (oi + o/ — \ji) .dp + \/G.Bia {a/ — i^) . dq 
VG'. sin (J. dq'=^ i/E . sin {^ft — oi) . dp + V G . sin tjf . dq 

and these equations, since they must be identical with 

dp'=adp + pdq 
dq''^ydp + hdq 

determine the coeflBcients a, j8, y, 8. We shall have 

"* AJ^'' sinoi' ' ^ ^E'' sinoi' 



_ IE sin(it-ai) 5. IG 



sina» 
sini/f 



G' sinoi' ' ^ G' sin 61' 

These four equations, taken in connection with the equations 

F _ . F' 



cos 01 = 



smoi 



=>/ 



VEQ' 
EG—F* 



cos OS r= 



EG 



sintu 



'=>! 



VE'G" 
E'G'—F'^ 



EG' 



may be written 

oi/(J?'<?'- 
/3i/(i?'<?'- 

8 i/(J?' <?'- 


-^' «)=l/i?<?'. sin (<» + «' - 
_^'8) -VEW. sin (^— 0,) 


Since by the substitutions 


dp'—adp+^dq, 
dg'=ydp + Sdq 



*) 



J 



GENERAL INVESTIGATIONS OF CURVED SURFACES 88 

the trinomial 

is transformed into 

Edf + 2Fdp.dq + ad^, 
we easily obtain 

and since, vice versa^ the latter trinomial must be transformed into the former by the 
substitution 

{aZ — fiy)dp = hdp'— fidq'y {ah — fiy)dq = — ydp' + adq\ 
we find 

EG—F^ 
E8^-2FyS + G'/= ^,^,_j,,, ^E' 

EG—F^ 
-EfiZ+F{aZ+Py)-Gay = j^^,_p„ ^F 

E^-2FaP + Ga'= ^g,_^,, G' 

22. 

From the general discussion of the preceding article we proceed to the very 
extended application in which, while keeping for p^ q their most general meaning, we 
take for p', q' the quantities denoted in Art. 15 by r, ^. We shall use r, ^ here 
also in such a way that, for any point whatever on the surface, r will be the shortest 
distance from a fixed point, and ^ the angle at this point between the first element 
of r and a fixed direction. We have thus 

E=\ F=% 6i'=90^ 
Let us set also 

VG'^m, 

so that any linear element whatever becomes equal to 

V{df^ + m^d<i?). 

Consequently, the four equations deduced in the preceding article for ci, )3, y, 8 give 

|/<7.co8^ = g^ (2) 



34 XASL FRIEDRIOH GATJSS 

V£!.Bixi{^-o>)=mM (3) 

dp 

l/(?.sml/f = »^.~? (4) 

dq 

But the last and the next to the last equations of the preceding article give 

(E.^-F.^].^-*=(F.'Ji-0.^\.^^ . (6) 

A dq dp' dq \ dq dp' dp 

From these equations must be determined the quantities r, <f>y ^ and (if need be) 
my as functions of p and q. Indeed, integration of equation (5) will give r ; r being 
found, integration of equation (6) will give <f> ; and one or other of equations (1), (2) 
will give ^ itself. Finally, m is obtained from one or other of equations (3), (4). 

The general integration of equations (5), (6) must necessarily introduce two arbi- 
trary functions. We shall easily understand what their meaning is, if we remem- 
ber that these equations are not limited to the case we are here considering, but are 
equally valid if r and <^ are taken in the more general sense of Art. 16, so that r is 
the length of the shortest line drawn normal to a fixed but arbitrary line, and ^ is 
an arbitrary function of the length of that part of the fixed line which is intercepted 
between any shortest line and an arbitrary fixed point. The general solution must 
embrace all this in a general way, and the arbitrary functions must go over into 
definite functions only when the arbitrary line and the arbitrary functions of its 
parts, which <f> must represent, are themselves defined. In our case an infinitely 
small circle may be taken, having its centre at the point from which the distances r 
are measured, and <f> will denote the parts themselves of this circle, divided by the 
radius. Whence it is easily seen that the equations (5), (6) are quite suflBcient for 
our case, provided that the functions which they leave undefined satisfy the condi- 
tion which r and <f> satisfy for the initial point and for points at an infinitely small 
distance from this point. 

Moreover, in regard to the integration itself of the equations (5), (6), we know 
that it can be reduced to the integration of ordinary diflPerential equations, which, how- 
ever, often happen to be so complicated that there is little to be gained by the reduc- 
tion. On the contrary, the development in series, which are abundantly sufficient for 
practical requirements, when only a finite portion of the surface is under considera- 
tion, presents no difficulty; and the formulae thus derived open a fruitful source for 



GENERAL INYE8TIGATI0NS OF CURVED SURFACES 85 

the solution of many important problems. But here we shall develop only a single 
example in order to show the nature of the method. 

23. 

We shall now consider the case where all the lines for which p is constant are 
shortest lines cutting orthogonally the line for which ^ = 0, which line we can regard 
as the axis of abscissas. Let A be the point for which r = 0, /) any point whatever 
on the axis of abscissas, AD =^p^ B any point whatever on the shortest line normal 
to ^ /) at Dy and BD=qy so that p can be regarded as the abscissa, q the ordinate 
of the point B. The abscissas we assume positive on the branch of the axis of 
abscissas to which ^ = corresponds, while we always regard r as positive. We take 
the ordinates positive in the region in which j> is measured between and 180°. 

By the theorem of Art. 16 we shall have 

61=90% ^=0, (7 = 1, 
and we shall set also 

\/E=n. 

Thus n will be a function of p, q^ such that for q = Q it must become equal to unity. 
The application of the formula of Art. 18 to our case shows that on any shortest 
line whaiever we must have 

de=-^' dp, 
dq 

where 6 denotes the angle between the element of this line and the element of the 
line for which q is constant. Now since the axis of abscissas is itself a shortest line, 
and since, for it, we have everywhere tf = 0, we see that for q = Q we must have 
everywhere 

dq 

Therefore we conclude that, if n is developed into a series in ascendiag powers of q, 
this series must have the following fonn: 

„ = 1 +/js + ^ j» + A J* + etc. 

where /, g, h, etc., will be functions of p, and we set 

f=fo +ffp +f"p* + etc. 
ff=ff° + g'p+g"j^+eto. 
h = h'> + h'p + h'Y + etc. 



86 KARL FREEBRICH GAUSS 

or 

n = l +r f+fpf+fY9^ + etc. 

+h^q^ + etc. etc. 

24. 

The equations of Art. 22 give, in our case, 

. , ar dr 9<A . , d6 

n&ui\jt = -^9 cosi/f = ^j —nco8\lt = m .^9 sin^ = m.-^, 

Va^/ ^ \ay ' 3^3^^ 3j9 dp 

By the aid of these equations, the fifth and sixth of which are contained in the others, 
series can be developed for r, <^, <^, w, or for any functions whatever of these quan- 
tities. We are going to establish here those series that are especially worthy of 
attention. 

Since for infinitely small values of jt?, q we must have 

the series for r* will begin with the terms J^^ + J^. We obtain the terms of higher 
order by the method of undetermined coeflBcients,* by means of the equation 

\n dp / \ oq f 
Thus we have 

[1] r^=/ + |/>Y + i/>V + (!/" - ^r')p'^ etc. 

Then we have, from the formula 

[2] rsm^=p-\rp^-\f'f^-{\f"^i^r')]^^ etc. 

* We have thought it useless to give the calculation here, which can be somewhat abridged by 
certain artifices. 



GENERAL mVESTIGATIONS OF OUHVED SURFACES 87 

and from the formula 

[3] rco8^ = y + f/V? + i/'/?+(|/"-A/°W «tc. 

These formulae give the angle ^ft. In like manner, for the calculation of the angle ^, 
series for r cos ^ and r sin <^ are very elegantly developed by means of the partial 
differential equations 

9.rcos<& A ' I * ± dd> 
il = n cos ^ . sm ^ — r sin ^ . 1^ 

dp dp 

9.rcos<4 • , • I 9<4 
J- = cos 4^ . cos i/f — r sm ^ . Ix 

dq dq 

ll!l^l5l = «8in^.sin^ + rcos<^.^ 

dp dp 

^•^^"'^=sin^.cosi/f+rcos<^.gi 
dq dq 

« cos Jf . -? + sin Jf . -? = 
^ dq^ ^ dp , 

A combination of these equations gives 

r sin li 9 . r cos <4 • , .9 . r cos <A . 

2: — r 4. r cos^ . 21 = r COS* 

n 9/? ^ g^r ^ 

rsinifc9.rsinA, , 9.rsin<& . , 
2: . — n + r cos^ • ^ = r sin 6 

n dp 9? 

From these two equations series for rcos^, rsin^^ are easily developed, whose first 
terms must evidently be jp, q respectively. The series are 

[4] roos<f>==p + ^rp^+^fff+{^r-:^f<»)ff etc. 

-iff'p'i'-ihff'fs' 

From a combination of equations [2] , [3] , [4] , [5] a series for r* cos (1^ + <^), may 
be derived, and, from this, dividing by the series [1], a series for cos(i^ + ^), from 



gives 



88 KARL FEIEDRICH GAUSS 

which may be found a series for the angle ^ + ^ itself. However, the same series 
can be obtained more elegantly in the following manner. By differentiating the first 
and second of the equations introduced at the beginning of this article, we obtain 

sin li h w cos Jf . — !? 4- sin Jf . -^ = 

and this combined with the equation 

n cos Jf . -? + sin li . -? = 
^ dg ^ dp 

r^in^ an r^ 

n dq n dp dq 

From this equation, by aid of the method of undetermined coefficients, we can easily 
derive the series for i/> + ^, if we observe that its first term must be ^ ir, the radius 
being taken equal to unity and 2 tt denoting the circumference of the circle, 

[6] ^+i>=h^-rM-^fp'q-{y"-\r')fi e^- 

- {h" - ir')pf 

It seems worth while also to develop the area of the triangle AB D into a series. 
For this development we may use the following conditional equation, which is easily 
derived from sufficiently obvious geometric considerations, and in which S denotes the 
required area: 



rsin^ dS . dS rsinii r , 

^•^ + rcosit.^ = Indq 

n dp ^ da n ^ 



dp ^ ' dq 

the integration beginning with q = 0. From this equation we obtain, by the method 
of undetermined coefficients, 

[7] S=^pq-^rp'q-i^f'p*q-{i^f"-i^r*)fq etc. 



GENERAL INVESTIGATIONS OP CURVED SURFACES 89 

25. 

From the formulse of the preceding article, which refer to a right triangle formed 
by shortest lines, we proceed to the general case. Let C be another point on the 
same shortest line D B, for which point p remains the same as for the point J?, and 
q'^ r', ^', i|/, /S* have the same meanings as q, r, ^, i/f, S have for the point B. There 
will thus be a triangle between the points A^ J?, C, whose angles we denote by 
Ay By Cy tho sldos opposite these angles by a, by Cy and the area by a. We represent 
the measure of curvature at the points Ay By C by a, j8, y respectively. And then 
supposing (which is permissible) that the quantities pyqy q — q' are positive, we shall 
have 

A=j>—j/y B=^y C=lT — y^y 

(i = q~ q\ b = r'y c? = r, <r = S— JSC. 

We shall first express the area <r by a series. By changing in [7] each of the 
quantities that refer to B into those that refer to Cy we obtain a formula for S\ 
Whence we have, exact to quantities of the sixth order, 

- -hf'P (6/+ 7 ?«+ 7 y ^' + 7 y'") 

This formula, by aid of series [2], namely, 

<? sin 5 =;> (1 - \r ^- \fpf- iic' ^- etc.) 

can be changed into the following: 

«r = iacsin5(l-i/°(p»-y«+y^+S/«) 

The measure of curvature for any point whatever of the surface becomes (by Art. 
19, where m, j?, q were what n, q, p are here) 

k = — - ^^ = 2/+6yg + 12Ag' + etc. 
~ n dg^~ 1 +f^ + etc. 

= - 2/- 6 y y - (12 A - 2/«) J* - etc. 
Therefore we have, when p, q refer to the point B, 

^ = _ 2/0 - 2/> - 6 y° (^ - 2/"/ - 6 /j9? - (12 A" - 2/°«) ^ - etc. 



4 



The consideration of the rectilinear triangle whose sides are equal to a, b, e is of 
great advantage. The angles of this triangle, which we shall denote by A*, -B*, (7*, 
differ from the angles of the triangle on the curved surface, namely, from Ay jS, C> 
by quantities of the second order; and it will be worth while to develop these differ- 
ences accurately. However, it will be sufficient to show the first steps in these more 
tedious than difficult calculations. 

Replacing in formulae [1], [4], [5] the quantities that refer to jB by those that 
refer to Cy we get formulae for r'*, r'cos^^', r'sin^'. Then the development of the 
expression 



40 KAHL PBIEDRICH GAUSS | 

Also 

y = -2fo- 2fp - 6 ff^'q' - 2/V -Qff'pq'- (12 A^ - 2/^*) /* - etc. 
a = -2r 

Introducing these measures of curvature into the expression for <r, we obtain the fol- 
lowing expression, exact to quantities of the sixth order (exclusive) : 

The same precision will remain, if for p, q, tf we substitute <? sin P, c cos B^ ecosB — a. 
This gives 

[8] <r = ^ac8in5(H-y^a(3a« + 4c«— 9aecosB) 

+ ^^(3a8 + 3c» — 12a<.cos5) 
+ Tky(^«' + 3c*— daeeoaB)) 

Since all expressions which refer to the line AD drawn normal to BC have disap- 
peared from this equation, we may permute among themselves the points A, By C and 
the expressions that refer to them. Therefore we shall have, with the same precision, j 

[9] <r=-ibcsmA(l + j^a{Sb* + S<^-12becoaA) 

-1-^^(3 i« + 4c«- 9becoaA) 
-h ^^TT y (4 J* + 3 c» - 9*ccos^)) 

[10] <T = ^abBmCCL + T^a{S(^ + ib'— 9abeosC) , 

+ Tk/8(4a» + 3«*- 9abcosC) 
+ Tky(3a* + 3a'-12aJcosC)) 4 

26. 



4 



i 



GENERAL INVESTIGATIONS OF CITEVED SURFACES 41 

r* + r**— {q — q')* — 2 r cos ^ . r* cos ^' — 2 r sin ^ . / sin ^' 
= b* + (*—(^ — 2beooBA 
= 2 dc (cos A* — cos A), 

combined with the development of the expression 

r sin ^ . r' cos ^' — r cos ^ . r' sin ^' = Jc sin il, 

gives the following formula: 

co8A*—coaA= — {q — q')pamAiif° + ^/'p + iff''{q + q') 

+ (iA'-iArr')(?* + ?/+n + etc.) 

From this we have, to quantities of the fifth order, 

A*-A = -^{q-q')piir + if/'p + y''{q + q') + ^fy 

+ ^9'pi9 + ^) + ih''{f + qq'+q") 

-T»V/'"(V+7ir'+12y?'+ 7?")) 
Combining this formnla with 

and with the values of the quantities a, fi, y found in the preceding article, we obtain, 
to quantities of the fifth order, 

[11] A*^A-<riia + ^fi + ^ + ^fy + iff'p{q + q') 

+ ih^(Sf-2qq'+Sq'*) 

+ ^r' (4/ -Uq' + Uqq'-U /«)) 
By precisely similar operations we derive 

[12] B*=B-<r(:^a + i^fi+^y + ^f'Y+^ff'p{2q + q-^ 

+ iA°(4y«-4yj' + 3y'«) 

[13] C*=C-a(:^a + ^fi + iy + ^fy + ^g'p(S + 2q') 

+ ih°{Bq'-iqq' + iq") 

From these formulae we deduce, since the sum A* + B"^ + Cf* is equal to two right 
angles, the excess of the sum A+B + C over two right angles, namely, 

[14] A+B + C=',r + <riia + ifi + iy + irjf^+Wpi9 + 9') 

+ (2A«-i/»«)(j«-y^+/«)) 

This last equation could also have been derived from formula [6]. 



42 / KASL FHIEDRICH GAUSS 



/ 



S7. 

If the curved surface is a sphere of radius B, we shall have 

or 

1 



A° = 



24 2J** 



Consequently, formula [14] becomes 



A+B+C=ir + ^ 
which is absolutely exact. But formulae [11], [12], [13] give 

or, with equal exactness. 

Neglecting quantities of the fourth order, we obtain from the above the well-known 
theorem first established by the illustrious Legendre. 

i 
28. 

Our general formulae, if we neglect terms of the fourth order, become extremely 
simple, namely : 

A* = A-^<r{2a+P + y) 
B*=B-^<rla + 2p + y) 



GENERAL INVESTIGATIONS OF CURVED SURFACES 48 

V. ...-' 

Thus to the angles A^ B^ C on bl non-spherical surface, unequal reductions must 
be applied, so that the sines of the changed angles become proportional to the sides 
opposite. The inequality, generally speaking, will be of the third order; but if the 
surface differs little from a sphere, the inequality will be of a higher order. Even in 
the greatest triangles on the earth's surface, whose angles it is possible to measure, 
the difference can always be regarded as insensible. Thus, e. g.j in the greatest of 
the triangles which we have measured in recent years, namely, that between the 
points Hohehagen, Brocken, Inselberg, where the excess of the sum of the angles was 
14'\85348, the calculation gave the following reductions to be applied to the angles : 

Hohehagen -4".95113 

Brocken . . . . . . — 4".95104 

Inselberg — 4".95131. 

29. 

We shall conclude this study by comparing the area of a triangle on a curved 
surface with the area of the rectilinear triangle whose sides are a, i, c. We shall 
denote the area of the latter by o-* ; hence 

<r* = ^ Jc? sin ii* = -J- ae? sin jB* = ^ a J sin 67* 

We have, to quantities of the fourth order, 

sinii*=sinii — -j^crcosil . (2a+j8 + y) 

or, with equal exactness, 

sinii = sinii*. (1 + ^Jccosii . (2 a+j8 + y)) 

Substituting this value in formula [9], we shall have, to quantities of the sixth order, 

cr = |i£?sinii*. (1 + T^a (3i^+ 3(?«-2*e? cos ii) 

+ ^^ (3 J«+ 4 (?«- 4 ic cos A) 

+^y(4j«+3o>-4i(?cosii)), 
or, with equal exactness, 

o.==o-*(l+T^a(a»+2*«+2(?«)+T^)8(2a«+i*+2c^)+T^y(2a»+2i>+ 

For the sphere this formula goes over into the following form: 

cr = <r* (1 + A:«K+ *'+ ^))- 



i 

44 EARL FBESDBICH GAUSS j 

It is easily verified that, with the same precision, the following formula may be taken 
instead of the above : I 



^Sl 



sin il • sin J? . sin 67 ^ 



sin Af^ . sin jB* . sin C^ \ 

K this formula is applied to triangles on non-spherical carved surfaces, the error, gen- 
erally speaking, will be of the fifth order, but will be insensible in all triangles such 
as may be measured on the earth's surface. 



( 



{ 



GAUSS'S ABSTRACT -- 45 



GAUSS'S ABSTRACT OF THE DISQUISITI0NE8 GBNERALES CIRCA 
SUPERFICIES CURVAS, PRESENTED TO THE ROYAL 

SOCIETY OF GOTTINGEN. 



(JOttingibchb gblbhbtb Anzbigbn. No, 177. Pages 1761-1768, 1827, November 5. 



On the 8tli of October, Hofirath Ghtuss presented to the Royal Society a paper : 

DtsquisUianes generdU% circa superficies curvas. 

Although geometers have given much attention to general investigations of carved 
surfaces and their results cover a significant portion of the domain of higher geometry, 
this subject is still so far from being exhausted, that it can well be said that, up to 
this time, but a small portion of an exceedingly fruitful field has been cultivated. 
Through the solution of the problem, to find all representations of a given surface upon 
another in which the smallest elements remain unchanged, the author sought some 
years ago to give a new phase to this study. The purpose of the present discussion 
is further to open up other new points of view and to develop some of the new truths 
which thus become accessible. We shall here give an account of those things which 
can be made intelligible in a few words. But we wish to remark at the outset that 
the new theorems as well as the presentations of new ideas, if the greatest generality 
is to be attained, are still partly in need of some limitations or closer determinations, 
which must be omitted here. 

In researches in which an infinity of directions of straight lines in space is con- 
cerned, it is advantageous to represent these directions by means of those points upon 
a fixed sphere, which are the end points of the radii drawn parallel to the lines. The 
centre and the radius of this auxiliary ^here are here quite arbitrary. The radius may 
be taken equal to unity. This procedure agrees fundamentally with that which is con- 
stantly employed in astronomy, where all directions are referred to a fictitious celestial 
sphere of infinite radius. Spherical trigonometry and certain other theorems, to which 
the author has added a new one of frequent application, then serve for the solution of 
the problems which the comparison of the various directions involved can present. 



46 GAUSS'S ABSTRACT 

If we represent the direction of the normal at each point of the curved surface by 
the corresponding point of the sphere, determined as above indicated, namely, in this 
way, to every point on the surface, let a point on the sphere correspond ; then, gener- 
ally speaking, to every line on the curved surface will correspond a line on the sphere, 
and to every part of the former surface will correspond a part of the latter. The less 
this part differs from a plane, the smaller will be the corresponding part on the sphere. 
It is, therefore, a very natural idea to use as the measure of the total curvature, 
which is to be assigned to a part of the curved surface, the area of the corresponding 
part of the sphere. For this reason the author calls this area the iniegral curvature of 
the corresponding part of the curved surface. Besides the magnitude of the part, there 
is also at the same time its position to be considered. And this position may be in 
the two parts similar or inverse, quite independently of the relation of their magni- 
tudes. The two cases can be distinguished by the positive or negative sign of the 
total curvature. This distinction has, however, a delGinite meaning only when the 
figures are regarded as upon definite sides of the two surfaces. The author regards 
the figure in the case of the sphere on the outside, and in the case of the curved sur- 
face on that side upon which we consider the normals erected. It follows then that 
the positive sign is taken in the case of convexo-convex or concavo-concave surfaces 
(which are not essentially different), and the negative in the case of concavo-convex 
surfaces. If the part of the curved surface in question consists of parts of these differ- 
ent sorts, still closer definition is necessary, which must be omitted here. 

The comparison of the areas of two corresponding parts of the curved surface and of 
the sphere leads now (in the same manner as, e. y., from the comparison of volume and 
mass springs the idea of density) to a new idea. The author designates as measure of 
curvature at a point of the curved surface the value of the fraction whose denominator is 
the area of the infinitely small part of the curved surface at this point and whose numer- 
ator is the area of the corresponding part of the surface of the auxiliary sphere, or the 
integral curvature of that element. It is clear that, according to the idea of the author, 
integral curvature and measure of curvature in the case of curved surfaces are analo- 
gous to what, in the case of curved Unes, are called respectively amplitude and curvar 
ture simply. He hesitates to apply to curved surfaces the latter expressions, which 
have been accepted more from custom than on account of fitness. Moreover, less 
depends upon the choice of words than upon this, that their introduction shall be justi- 
fied by pregnant theorems. 

The solution of the problem, to find the measure of curvature at any point of a curved 
surface, appears in different forms according to the manner in which the nature of the 
curved surface is given. When the points in space, in general, are distinguished by 



GAUSS'S ABSTRACT ^ 47 

three rectangular coordinates, the simplest method is to express one coordinate as a func- 
tion of the other two. In this way we obtain the simplest expression for the measure of 
curvature. But, at the same time, there arises a remarkable relation between this 
measure of curvature and the curvatures of the curves formed by the intersections of 
the curved surface with planes normal to it. Euler, as is well known, first showed 
that two of these cutting planes which intersect each other at right angles have this 
property, that in one is found the greatest and in the other the smallest radius of cun- 
vature ; or, more correctly, that in them the two extreme curvatures are found. It will 
follow then from the above mentioned expression for the measure of curvature that this 
will be equal to a fraction whose numerator is unity and whose denominator is the product 
of the extreme radii of curvature. The expression for the measure of curvature will be 
less simple, if the nature of the curved surface is determined by an equation in x, y, z. 
And it will become still more complex, if the nature of the curved surface is given so that 
Xy y, z are expressed in the form of functions of two new variables p, q. In this last case 
the expression involves fifteen elements, namely, the partial differential coefficients of the 
first and second orders of a?, y, z with respect to p and q. But it is less important in itself 
than for the reason that it facilitates the transition to another expression, which must be 
classed with the most remarkable theorems of this study. If the nature of the curved 
surface be expressed by this method, the general expression for any linear element upon 
it, or for Vids? + rf/ + rf^), has the form \/{Edp^ + 2 Fdp .dq + G df), where JE, F, G 
are again functions of p and q. The new expression for the measure of curvature men- 
tioned above contains merely these magnitudes and their partial differential coefficients 
of the first and second order. Therefore we notice that, in order to determine the 
measure of curvature, it is necessary to know only the general expression for a linear 
element ; the expressions for the coordinates x, y, z are not required. A direct result 
from this is the remarkable theorem : If a curved surface, or a part of it, can be devel- 
oped upon another surface, the measure of curvature at every point remains unchanged 
after the development. In particular, it follows from this further: Upon a curved 
surface that can be developed upon a plane, the measure of curvature is everywhere 
equal to zero. From this we derive at once the characteristic equation of surfaces 
developable upon a plane, namely, 

ds? *a/ \dx.dyi """' 

when z is regarded as a function of x and y. This equation has been known for some 
time, but according to the author's judgment it has not been established previously 
with the necessary rigor. 



48 GAUSS'S ABSTRACT 

These theorems lead to the consideration of the theory of curved surfaces from a 
new point of view, where a wide and still wholly uncultivated field is open to investi- 
gation. If we consider surfaces not as boundaries of bodies, but as bodies of which 
one dimension vanishes, and if at the same time we conceive them as flexible but not 
extensible, we see that two essentially different relations must be distinguished, namely, 
on the one hand, those that presuppose a definite form of the surface in space ; on the 
other hand, those that are independent of the various forms which the surface may 
assume. This discussion is concerned with the latter. In accordance with what has 
been said, the measure of curvature beloi^ to this case. But it is easily seen that 
the consideration of figures constructed upon the surface, their angles, their areas and 
their integral curvatures, the joining of the points by means of shortest lines, and the 
like, also belong to this case. All such investigations must start from this, that the 
very nature of the curved surface is given by means of the expression of any linear 
element in the form \/{Edj;^+2Fdp .dq + Gdq^). The author has embodied in the 
present treatise a portion of his investigations in this field, made several years ago, 
while he limits himself to such as are not too remote for an introduction, and may, to 
some extent, be generally helpful in many further investigations. In our abstract, we 
must limit ourselves still more, and be content with citing only a few of them as 
types. The following theorems may serve for this purpose. 

If upon a curved surface a system of infinitely many shortest lines of equal lengths 
be drawn from one initial point, then wiU the line going through the end points of 
these shortest lines cut each of them at right angles. If at every point of an arbitrary 
line on a curved surface shortest lines of equal lengths be drawn at right angles to this 
line, then will all these shortest lines be perpendicular also to the line which joins their 
other end points. Both these theorems, of which the latter can be regarded as a gen- 
eralization of the former, will be demonstrated both analytically and by simple geomet- 
rical considerations. The excess of the sum of the angles of a triangle formed by shortest lines 
over two right angles is equal to the total curvature of the triangle. It will be assumed here 
that that angle (57° 17' 45") to which an arc equal to the radius of the sphere corresponds 
will be taken as the unit for the angles, and that for the unit of total curvature will be 
taken a part of the spherical surface, the area of which is a square whose side is equal to 
the radius of the sphere. Evidently we can express this important theorem thus also : 
the excess over two right angles of the angles of a triangle formed by shortest lines is to 
eight right angles as the part of the surface of the auxiliary sphere, which corresponds 
to it as its integral curvature, is to the whole surface of the sphere. In general, the 
excess over 2 n — 4 right angles of the angles of a polygon of n sides, if these are 
shortest lines, will be equal to the integral curvature of the polygon. 



GAUSSES ABSTRACT . 49 

The general investigations developed in this treatise will, in the conclusion, be applied 
to the theory of triangles of shortest lines, of which we shall introduce only a couple of 
important theorems. If a^ by c be the sides of such a triangle (they will be regarded as 
magnitudes of the first order) ; A^B^C the angles opposite ; a, /8, y the measures of 
curvature at the angular points ; o- the area of the triangle, then, to magnitudes of the 
fourth order, \ (a +/8 + y) o- is the excess of the sum -4 +i? + (7 over two right angles. 
Further, with the same degree of exactness, the angles of a plane rectilinear triangle 
whose sides are a, d, c, are respectively 

^-l^(2a+/8 + y)cr 

i?-^(a+2/8 + y)<r 

67-^(a+/8+2y)<r. 

We see immediately that this last theorem is a generalization of the familiar theorem first 
established by Legendre. By means of this theorem we obtain the angles of a plane 
triangle, correct to magnitudes of the fourth order, if we diminish each angle of the cor- 
responding spherical triangle by one-third of the spherical excess. In the case of non- 
spherical surfaces, we must apply unequal reductions to the angles, and this inequality, 
generally speaking, is a magnitude of the third order. However, even if the whole sur- 
face differs only a little from the spherical form, it will still involve also a factor denoting 
the degree of the deviation from the spherical form. It is unquestionably important for 
the higher geodesy that we be able to calculate the inequalities of those reductions and 
thereby obtain the thorough conviction that, for all measurable triangles on the surface 
of the earth, they are to be regarded as quite insensible. So it is, for example, in the 
case of the greatest triangle of the triangulation carried out by the author. The greatest 
side of this triangle is almost fifteen geographical "^ miles, and the excess of the sum 
of its three angles over two right angles amounts almost to fifteen seconds. The three 
reductions of the angles of the plane triangle are 4".95113, 4".95104, 4".95131. Besides, 
the author also developed the missing terms of the fourth order in the above expres- 
sions. Those for the sphere possess a very simple form. However, in the case of 
measurable triangles upon the earth's surface, they are quite insensible. And in the 
example here introduced they would have diminished the first reduction by only two 
units in the fifkh decimal place and increased the third by the same amount. 



* This German geographical mile is four minutes of arc at the equator, namely, 7.42 kilome- 
ters, and is equal to about 4.6 English statute miles. [Translators.] 



/^ 



\ 



1 



NOTES 61 



NOTES. 

Art. 1, p. 3, 1. 3. Oauss got the idea of using the auxiliary sphere from astron- 
omy. Cf. Oauss's Abstract, p. 45. 

Art. % p. 3, 1. 2 fr. hot. In the Latin text 9itw is used for the direction or 
orientation of a plane, the position of a plane, the direction of a line, and the posi- 
tion of a point. 

Art. 2, p. 4, 1. 14. In the Latin texts the notation 

cos (1) D + cos (2) n + cos (3) U = \ 

is used. This is replaced in the translations (except B5klen's) by the more recent 
notation 

cos* (1) L + cos* (2) L + cos* (3) Z = 1. 

Art. 2, p. 4, 1. 3 fr. bot. This stands in the original and in Liouville's reprint, 

cos A (cos ^ sin ^ — sin ^ cos ^) (cos if' sin V — sin If' 9in f). 

Art. 2, pp. 4-6. Theorem YI is original with Oauss, as is also the method of 
deriving YII. The following figures show the points and lines of Theorems YI and 
YII: 





Art. 3, p. 6. The geometric condition here stated, that the curvature be continu- 
ous for each point of the surface, or part of the surface, considered is equivalent to 
the analytic condition that the first and second derivatives of the function or frino- 
tions defining the surface be finite and continuous for all points of the surface, or 
part of the surface, considered. 

Art. 4, p. 7, 1. 20. In the Latin texts the notation XX for JT*, etc., is used. 



52 NOTES 

Art. 4, p. 7. " The second method of representing a surface (the expression of 
the coordinates by means of two auiiliary variables) was first used by Gauss for 
arbitrary surfaces in the case of the problem of conformal mapping. [Astronomische 
Abhandlungen, edited by H. C. Schumacher, vol. Ill, Altona, 1825; Gauss, Werke, 
vol. IV, p. 189 ; reprinted in vol. 55 of Ostwald's Klassiker. — Cf. also Gauss, Theoria 
attractionis corporum sphaer, ellipt., Comment. Gott. II, 1813 ; Gauss, Werke, vol. V, 
p. 10.] Here he applies this representation for the first time to the determination of 
the direction of the surface normal, and later also to the study of curvature and of 
geodetic lines. The geometrical significance of the variables p^ q is discussed more fully 
in Art. 17. This method of representation forms the source of many new theorems, 
of which these are particularly worthy of mention: the corollary, that the measure of 
curvature remains unchanged by the bending of the surface (Art. 11, 12) ; the theorems 
of Art. 15, 16 concerning geodetic lines ; the theorem of Art. 20 ; and, finally, the 
results derived in the conclusion, which refer a geodetic triangle to the rectilinear trian- 
gle whose sides are of the same length." [Wangerin.] 

Art. 5, p. 8. "To decide the question, which of tiie two systems of values found 
in Art. 4 for JT, JT, Z belong to the normal directed outwards, which to the normal 
directed inwards, we need only to apply the theorem of Art. 2 (VII), provided we use 
the second method of representing the surface. If, on the contrary, the surface is 
defined by the equation between the coordinates Tr= 0, then the following simpler con- 
considerations lead to the answer. We draw the line rfor from the point A towards 
the outer side, then, if rfar, rfy, dz are the projections of rfo-, we have 

Pdx-\- qdy-^Rdz>^. 

On the other hand, if the angle between o- and the normal taken outward is acute, 

then 

dx dv dz 

ac ac ac 

This condition, since da- \s positive, must be combined with the preceding, if the first 
solution is taken for JT, F, Z, This result is obtained in a similar way, if the. sur- 
face is analytically defined by the third method." [Wangerin.] 

Art. 6, p. 10, 1. 4. The definition of measure of curvature here given is the one 
generally used. But Sophie Germain defined as a measure of curvature at a point of 
a surface the sum of the reciprocals of the principal radii of curvature at that point, 
or double the so-called mean curvature. Cf. Crelle's Joum. fiir Math., vol. VII. 
Casorati defined as a measure of curvature one-half the sum of the squares of the 
reciprocals of the principal radii of curvature at a point of the surface. Cf. Rend, 
del R. Istituto Lombardo, ser. 2, vol. 22, 1889 j Acta Mathem. vol. XIV, p. 95, 1890. 



NOTES 68 

Art. 6, p. 11, L 21. Gauss did not carry out his intention of studying the most 
general cases of figures mapped on the sphere. 

Art. 7, p. 11, 1. 31. "That the consideration ' of a surface element which has the 
form of a triangle can be used in the calculation of the measure of curvature, follows 
from this fact that, according to the formula developed on page 12, k is independent 
of the magnitudes rfar, rfy, 8ar, 8y, and that, consequently, k has the same value for 
every infinitely small triangle, at the same point of the surface, therefore also for sur- 
face elements of any form whatever lying at that point." [Wangerin.] 

Art. 7, p. 12, 1. 20. The notation in the Latin text for the partial derivatives : 

dX dX ^ 
li' 1^' ^*^-' 

has been replaced throughout by the more recent notation: 

dX dX ^ 
-^ — ? "5 — 9 etc. 

Art. 7, p. 13, 1. 16. This formula, as it stands in the original and in Liouville's 

reprint, is 

dY^-Z^tudt-Z'^i^X^- f) du. 

The incorrect sign in the second member has been corrected in the reprint in Gauss, 
Werke, vol. IV, and in the translations. 

Art. 8, p. 15, .1. 3. Euler's work here referred to is found in Mem. de TAcad, 
de Berlin, vol. XVI, 1760. 

Art. 10, p. 18, 11. 8, 9, 10. Instead of /?, /)', D" as here defined, the Italian 
geometets have introduced magnitudes denoted by the same letters and equal, in 
Gauss's notation, to 

D D' D" 

'viHG-Fy V{EG-F^)' V^EG-F") 
respectively. 

Art. 11, p. 19, 11. 4, 6, fr. bot. In the original and in Liouville's reprint, two of 
these formulae are incorrectly given : 

dF „ , dF \ dE 

dq ' dq 2 dq 

The proper corrections have been ma&e in Gauss, Werke, vol. IV, and in the trans- 
lations. 

Art. 13, p. 21, 1. 20. Gauss published nothing further on the properties of devel- 
opable surfaces. 



M NOTES 

Art. 14, p. 22, 1. 8. The transformation is easily made by means of integration 
by parts. 

Art. 17, p. 26. If we go from the point p, q to the point (jp + dpj q\ and if the 
Cartesian coordinates of the first point are x, y, Zy and of the second rr + dx^ y + dy^ 
e + da; with ds the linear element between the two points, then the direction cosines 
of ds are 

dx A dtf dz 

cosa = -=-, cosj8 = -5^, cosy = .^-. 

as ds ds 

Since we assume here j^= Constant or dq = Oy we have also 

dx dy dz 

^^""d^'^Py ^y^d^'^Py ^^^d^'^Py ds^±V£.dp. 

If dp is positiye, the change ds will be taken in the positive direction. Therefore 
ds = v'E.dpy 

1 dx A 1 3y 1 dz 

In like manner, along the line p = Constant, if cos a', cos ^S', cos y are the direction 
cosines, we obtain 

1 3:p 1 dy 1 dz 

cosa' = :pr^-g^, co8^ = pr^-g^, cos y = pr^ • g^. 

And since 

co8«> — co8acosa'+cos/8oo8/3'+cosyco8y, 
F 

From this follows 

V{EQ-F^ 

And the area of the quadrilateral formed by the lines j?, j9 + dp, q^q + dq is 

da=V{EG-F^).dp.dq. 

Art. 21, p. 33, 1. 12. In the original, in Liouville's reprint, in the two French 
translations, and in B(5klen's translation, the next to the last formula of this article 
is written 



NOTES 



65 



The proper correction in sign has been made in Ckiuss, Werke, voL IV, and in Wan- 
gerin's translation. 

Art. 23, p. 36, 1. 13 fr. hot. In the Latin texts and in Roger's and Bdklen's 
translations this formula has a minus sign on the right hand side. The correction in 
sign has been made in Abadie's and Wangerin's translations. 

Art. 23, p. 35. The figure below represents the lines and angles mentioned in 
this and the following articles : 




Art. 24, p. 36. Deriyation of formula [1]. 

Let r"=y + f+It^+It^+It^+It^+ etc. 

where R^ is the aggregate of all the terms of the third degree in p and q^ R^ of all 
the terms of the fourtii degree, etc. Then by differentiating, squaring, and omitting 
terms above the sixth degree, we obtain 



dp 



dp' \dp 



dp 



+ 4^^ + 4^^. + 2y?8?^ + 2 



and 



dp 



dp dp dp 



dp 

dR^d^ 

dp dp^ 



(B*-«'+e)'+fi!')'+*/^+*.'^ 



dp 



dq 



Bq 



dq 



dq 



^*^^ dq^*^ dq^^ dq dq ^ ^ dq dq 



66 NOTES 

Hence we have 



mhm)'-^" 



dp f \ dq 



\^ dp ^ dq ^ ^ dp dp 2 ^q Qg / 

since^ according to a familiar theorem for homogeneous functions , 



3 Xig 9 Xijj 



■Pl7 + ?-a7-3^«'«*«- 



By dividing unity by the square of the value of n, given at the end of Art. 23, and 
omitting terms above the fourth degree, we have 

1 
1 J = 2/° y« + 2/'pf + 2^ V - 3/°*?* + 2/"pY + 2ff'pf + 2 A» q*. 

This, multiplied by the last equation but one of the preceding page, on rejecting terms 
above the sixth degree, becomes 

(3 7? V* 



dE, 



+ ^rpfdp 



Therefore, since from the fifth equation of Art. 24 : 



(f^M^)"-*-=(i-^)(W' 



NOTES 67 

we have 

Whence, by the method of undetermined coefficients, we find 
And therefore we have 

[1] f»=^f+ irf^ + i/>Y + (ir- iV/'")i'Y + etc. 

This method for deriving formula [1] is taken from Wangerin. 
Art. 24, p. 36. Derivation of formula [2]. 

By taking one-half the reciprocal of the series for n given in Art. 23, p. 36, we 
obtain 

And by differentiating formula [1] with respect to J9, we obtain 

Therefore, since 

we have, by multiplying together the two series above. 



68 NOTES 

Art. 24, p. 37. Derivation of formula [3]. 

By differentiating [1] on page 57 with respect to q^ we find 

Therefore we have, since 

[3] r cos ,^ = y + \rfq + \f'fq + {y - ■hr')P*q + etc. 

Art. 24, p. 37. Derivation of formula [4]. 

Since r cos ^ becomes equal to p for infinitely small values of p and ^, the series 
for r cos ^ must begin with jo. Hence we set 

(1) r cos ^ = j9 +-B,+53+^4+55+ etc. 

Then, by differentiating, we obtain 

(3) ^^=f + t + t + t + «*«- 

By dividing [2] p. 57 by n on page 36, we obtain 

T sm 1^ 

(4) —^ =p - irp^ - \f'ft - (*/" + i^nft - etc. 

Multiplying (2) by (4), we have 



-*/>v - */>VTf - (*A° -*!/'•) /^^ 



3^ 

3j» 



|y>?" -y^p^j^"-^^' 



NOTES 69 



Multiplying (3) by [3] p. 58, we have 

(6) rco8^.-L_W = ^__. + ^_8 + ^__i +^_. +yygjf 

Since 

r sin ilr 9 (r cos 6) , 9 (r cos A) 

—a 3^ + rcoB^I,' 9^=rco8^, 

we have, by setting (1) equal to the sum of (5) and (6), 
p + Rt+ R,+ R^+ R^+ etc. 

+/-^-irp^-irp/-^-irp/-^ +/^ 



go 



from which we find 



Therefore we have finally 

[4] r cos ^ =i> + !/'/»?* + A/y ?« + (Ar - A/")/'V + etc. 

Art. 24, p, 37. Derivation of formula [5]. 

Again, since r sin ^ becomes equal to q for infinitely small values of p and q^ 
we set 

(1) r sin ^ = g' + Jf,+i?3+i?^+-B^+ etc. 



60 NOTES 

Then we have by differentiation 

^ ^ dp dp dp dp dp 

^ ^ dq dq dq dq dq 

MultiplTing (4) p. 58 by this (2), we obtain 

Likewise from (3) and [3] p. 68, we obtain 

/e\ . 3 (rain A) , dR^ dR^ . dR. . dR, 

(6) rcosi^. %y "^' -g + gj^' + gif + giTj + giy +{if"-^np'q 

dR d R 

dR 

+ y'fi + \ffi 77 + (i A" - ^^nft 



Since 



dR 



rsinJf 3(rsmA) . . 9 (r sin A) . , 

n 3jt? ^ dq ^* 



by setting (1) equal to the sum of (4) and (6), we have 
q +i?,+i?3+i?^+i?5+ etc. 

3 ff 3 7? 



NOTES 61 

from which we find 

^.= - (A/" - A/"')/? - ^9'p'q' - (i A° +**/")/?•. 

Therefore, substituting these values in (1), we have 

[5] r8in^ = y-i/>/y-i/yy -(iV/"-i^/°*)/?-etc. 

-y^ff- ^ff'p'9' 

Art. 24, p. 38. Derivation of formula [6]. 
Differentiating n on page 36 with respect to q, we obtain 

(1) 1^ = 2/> y + 2/> y + 2/y ? + etc. 

+ 4^*^* + etc., etc. 
and hence, multiplying this series by (4) on page 58, we find 

(2) ^ • If = 2/>;,y + 2/yq + 2/V ? + 3 ff'py + etc. 

IT 

For infinitely small values of r, ^ + ^ = k? as is evident from the figure on page 
65. Hence we set 

Then we shall have, by differentiation, 

/gx !(!Hl^=i^i + !^« + !^ +!:?«+ etc 

^ ^ dp dp dp dp dp 

Therefore, multiplying (4) on page 58 by (3), we find 

rsinj, 3(^ + <^) ^ 3^ 8^ 9^3 +«^* + etc 

^ ^ n 3jo ^ dp ^ dp ^ dp ^ dp 



R J-f ^ 



dp ^J -fi 9jt> 

if pr-3j 

-W'P^-s^' 



And since 



62 NOTES 

and, multiplying [S] on page 68 by (4), we find 

(6) reoB^f,- 3^ "^af +3'T7 ^^^ 3^ +^3^ +«*<'• 

op 

we shall have, bj adding (2), (5), and (6), 

9-di+Plf +3^>,« +Sff'py +g-^ 

ti R ti R 7i R 

+irfq-^-\rff^ +etc. 

From this equation we find 
Therefore we haye finallj 

[6] ^^<^=l-rpq-U'fq-iy"-\r)fq-^^- 

-{h--\np^' 



NOTES 68 

Art. 24, p. 38, 1. 19. The differential equation from which formula [7] follows 
is derived in the following manner. In the figure on page 55, prolong AD to jy, 
making Diy = dp, through 2>' perpendicular to AD' draw a geodesic line, which will 
cut AB in B'. Finally, take D'B" = DB, so that BB" is perpendicukr to B'D*. 
Then, if by ABD we mean the area of the triangle ABD, 

dS ,. AB'D'-ABD ,. BDD'B' ,. BDUB" ,. DU 
g^ = hm -^^, = \xm ^^, =lim-^^ — ^TW' 

since the surface BDD'B" differs from BDD'B' only by an infinitesimal of the 
second order. And since 



r BDD'B" r 

BDD'B" ==dpjn dq, or lim — -^^ — =J n rfy. 



and since, further, 



consequently 



Therefore also 



.. DD' dp 
^BW^d?' 



dp dr ^ dq'dr dr J"'*^' 



dr dr 
Finally, from the values for g-« r- given at the beginning of Art. 24, p. 36, we have 

9o 1 . dq 

so that we have 

[Wangerin.] 

Art. 24, p. 38. Derivation of formula [7] . 

For infinitely small values of p and q^ the area of the triangle ABC becomes 
equal to \ pq* The series for this area, which is denoted bj 8y must therefore begin 
with \ pq, or if,. Hence we put 

8= R^ + R^+R^+R^+R^+ etc. 



64 NOTES 

By differentiatiiig, we obtain 

o/S o Jim I o Ji^ o Ji , o Urn o jtC» 

n\ _ = » + » -|_ ^ -|_ g -|_ g -j_ g^^ 

^ ^ dp dp dp dp dp dp *' 

^ ^ dq dg 9y 9y dq dq *' 

and therefore, by multiplying (4) on page 58 by (1), we obtain 

^ ^ n dp ■'^ dp -^ dp J^ dp ^ dp ^ dp 

dR 

-(♦/" + A/" V«"-3^ 

op 

- (I A" -If /">?*— 



3j9' 



and multiplying [3] on page 58 by (2), we obtain 

(4) ,c<«*f=,f + /^ + ,f +,f +,^ + etc. 



NOTES 65 

Integrating n on page 36 with respect to g, we find 
(5) Sndq = q + irf + i/'pf+i/'yf+^^' 

+ iy''y*+i//'?*+etc. 

+ |A"»j*+ etc. etc. 

Multiplying (4) on page 58 by (5), we find 

(6) t^ .fn dq =pq -rp^ - wYf - mr + i^nf^ - etc. 

Since 

we obtain, by setting (6) equal to the sum of (3) and (4), 

pq -tpf ~Wft -(il/" + A/°*)/?'-etc. 

-(*A°-H/"')i'?" 

+^'^+^^•+^17 -i/^i'^^'+.f' -dr+Arw.^' 



From this equation we find 

R^=hP9> ^8=o» ^4^ — ^rpf-^r^q. 



66 NOTES 

Therefore we have 

-(TVA'-TAr/")i'?». 

Art. 26, p. 39, 1. 17. 3j9» + 4 ^ + 4 y^ + 4 ^« is replaced by 3/ + 4 j« + 4^«. 
This error appears in all the reprints and translations (except Wangerin's). 

Art. 25, p. 40, 1. 8. 3 / — 2 y« + y^ + 4 yy* is replaced by 3 / — 2 j* + yj-' 
+ i g**. This correction is noted in all the translations, and in Liouville's reprint. 

Art. 26, p. 40. Derivation of formulae [8] , [9] , [10] . 

By priming the q's in [7] we obtain at once a> series for S'. Then, since 
<r = S—S', we have 

-^rp i^-r)-Thf'fi^-r)-^9''pi^-^% 

correct to terms of the sixth degree. 

This expression may be written as follows : 

a = ^p{q-q')i\- ^ ril^ + 9^ + 99' + 9") 

-tV/'i'(6/ + 7?» + 7yy' + 7n 

-^9''i9 + 9')(.^f +4j« +4^), 
or, after factoring, 

(1) <r^ipi9-9')(>-\r9'-\rP9'-y''f)0--ir{l^-9' + 99' + ^') 

-)rV/>(6/-8?'+7?y'+7s^)-3j«jy°(3/y+3/y'-6j«+4y«y'+4y/«+4^»)). 

The last factor on the right in (1) can be written thus: 
<l-T^/'(4/) -Thn^l^) -Thfpi^99')-Thri^l^) -ThfPi99l 

+ Thr (2 f) + Thr (6 ?») - Thfp (3 ^») + Thr (2 f) - jh/'p (^ ?") 
-Thr{^99')-Thri^99')-Thff''9i^p') -Thrw) —Thff''9'{^i^ 

-Thri^9'*)-Thri^9")+Thff'9{^f) -ThrHq") +Th9''9'i29') 

- Thfp (3;^) - Thff" 9(^99')- Thfp (3/) - xlirr 9'(S9') 
+ Thfp{^9')-Th9'9(^9") +Thfpi^f) -TlTryYC*?")). 

We know, farther, that 

1 3*n 
*=""»-97 = -2/-^^?-(12A-2/')?'~etc., 



NOTES 67 

y = y° +/;> + /'/+ etc-, 

Hence, sabstitatmg these values for /, ff, and h m k, we have at B where i = fi, 
correct to terms of the third degree, 

/8 = -2/'-2/';>-6y°y-2/'V-6/i>y-(12A»-2/°«)^. 

Likewise, remembering that q becomes g' at (7, and that both p and q vanish at A, 
we have 

y = _2/>-2/>-6^°3r'-2/V-6/i>?'-(12A°-2/°«)^«, 
o = -2/>. 

And since c sin S = r sin ^, 

csin5=i»(l-i/'^-i/i>j«-i^V-etc.). 

Now, if we substitute in (1) <; sin j?, a, /8, y for the series which they represent, 
and a for j* — q', we obtain (still correct to terms of the sixth degree) 

<r = ^ac sinS (1 + y^ o (4/- 2 j» + 3 y/ + 3 y") 

+ Tk/8(3/-6y«+6yy'+30 
+ Tky(3/.-2j«+ qq'+4:q^y 

And if in this equation we replace p, g, q' by <? sin ^, c cos -B, <? cos -B — a, respect- 
iyelj, we shall have 

[8] cr = ia(?sin5(l + y|^a(3a*+4c*— 9accosj5) 

+ Y^)8(3 a^ + 3 c*- 12 a(? cos 5) 
+ y|^y(4a»+3c*- 9a(?co8j5)). 

By writing for B, aLy fi, a in [8], -4, )8, a, b respectively, we obtain at once 
formula [9]. Likewise by writing for -B, )8, y, c in [8], C, y, )8, i respectively, we 
obtain formula [10]. Formulse [9] and [10] can, of course, also be derived by the 
method used to derive [8]. 

Art. 26, p. 41, 1. 11. The right hand side of this equation should have the pos- 
itive sign. All the editions prior to Wangerin's have the incorrect sign. 

Art. 26, p. 41. Derivation of formula [11]. 

We have 

(1) f* + r'*-(y-/)*-2rcos^.r'cos^'-2rsin^.r'sinf 

= J« + c* - a« - 2 J(? cos (<^ - f ) 

= 2 be (cos A* — cos -4), 
since J* + c* — a' == 2 ic cos A* and cos (^ -* ^') = cos A. 



68 NOTES 

By priming the q*B in formulae [1], [4], [5] we obtain at once series for r", 
r' cos ^'y r' sin ^'. Hence we have series for all the terms in the above expression, 
and also for the terms in the expression: 

(2) rsin^.r'oos^' — rcos^ .r'sin^' = Jcsinil, 
namely, 

(3) »* = / + i/°/?' + i/>V+(i/"-:iV/'V?'+etc. 

(4) r'» =y + \rfr + yy 9" + d/' - -^np'r + etc. 

(5) -{q-g'Y- f+2gg'-g'\ 

(6) 2rco8<^=2;, + irp^ + H/y?'+ (A/" -if /")/?" + etc. 

(7) r' cos f =/; + !/»/» ^ + A/y ?" + (A/" - A/"')/?^ + etc. 

(8) 2rsin^ = 2y-|/'/y-|/yy -(A/"-H/°*)/?-etc. 

(9) r'sinf = gr'-|/';,«y'-|/yy' -(TV/"-iftr/*V?'-etc. 

By adding (3), (4), and (5), we obtain 
(10)r»+r«-(y-j')'=2/+|/»/(j«+0+i//(?'+?'*)+a/'-iV/°V(?'+?'*)+eto. 

On multiplying (6) by (7), we obtain 

(11) 2 r cos ^ . r' cos ^' 

= 2/+ ir/(j» + n + !/'/(?* + ?'•) + (ir-H/")/(j* + ?") + etc. 



NOTES 69 

and multiplying (8) by (9), we obtain 

(12) 2r8in^.r'sinf 

-^^4-\rfi^-\f'fi^ -(*/"-!*/'>*??' -etc. 

Hence by adding (11) and (12), we have 

(13) 2£ccoSil 

= 2^ + 4/>/(^ + ^«) + i/y(5j»-4yy'+51y'«)-Vt/°V(2j»+2/»-3y/) + etc. 

- tV/° '/ (14 ?* + 14 y'* + 13^ / + 18 yy'» - 40 ^q") 
+ TV//(7?'+7r-3j«y'-3yy«) 
+ i/V(3?'+3^«-2y^) 

+ JAV(2y* + 2^*-?»/-y^»). 

Therefore we have, by subtracting (13) from (10), 

2 he (cos A* — cos 4) 

= -*/°/(?' + /*-2??')-i/'/(^ + ?"-2y/)+A/>V(?« + jr'»-2yy')-etc. 

+ tV/°V(7?* + 7 ?'*+ 13 ^y' + 13 y^ - 40 ^q'*) 

which we can write thus : 

(14) 2 he (cos .I* - cos il) = - 2 / (y - /)« (i/- + i/> + i^° (^ + j^) + ^^r/'y 

+ iA° (?«+?/+ n + ^g'pia^ q') 
- Ary - T,V/°M7 ?• + 7 ^« + 27 y^)), 

correct to terms of the seventh degree. 

If we multiply (7) by [5] on page 37, we obtain 

(15) r sin ^ . r' cos j>' 

=pg + irP99"+ A/'/ 99" + (A/" - A/")/ 99" - etc. 
-iri^9 +^if''P99" +^3^1^99" 

-\f¥'9 +{ih--i^nP9r 

-i^>v -^r'p'99" 

-iThfr-^r')i^9 

-i^g'pW 



70 NOTES 

And multiplying (9) by formula [4] on page 37, we obtain 

(16) r cos ^ . r' sin ^' 

+ ^/Y ?• ?' - a A° + i*/°*)/ r 

Therefore we have, by sabtracting (16) from (15), 

(17) be Bin A 

correct to terms of the seventh degree. 

Let -4* — il = {, whence A*=A+ Cy C being a magnitude of the second order. 
Hence we have, expanding sin { and cos {, and rejecting powers of { above the second, 



cos 
or 



il*=cosil . ^1 — 2") "" siuil . i, 



cos A 
cos il* — cos -4 = 2 i — sin il . { ; 

or, multiplying both members of this equation by 2ley 

(18) 2 be (cos il* — cos -4) = — i c cos il . {* — 2 J c sin il . i. 

Further, let l^=B^+Ii^+B^+etc.f where the B's have the same meaning as before. 
If now we substitute in (18) for its various terms the series derived above, we shaU 
have, on rejecting terms above the sixth degree, 

(/ + ??') R>'+ ^P (? - y') (1 - ir (f^ + 2 gg')) iB,+B,+B;) 
+ i AM?" + y?' + n - tjV/"' (12/ + 7 ^ + 7 ?"+ 27 ?/)). 



NOTES 71 

Equating terns of like powers, and solving for R^ R„ R„ we find 

-^r' (7/ + 7 ?« + 7 /« + 12 qq')y 
Therefore we have 

A*-A=p{q-g')iir + ifp +:y' (?+?') +tV/"/ 

correct terms of the fifth degree. 

This equation may be written as follows : 

But, since 

the above equation becomes 

A*=^A-ai-ir-^fp-y'{9 + g')-\fy-^ff'p(S! + 9') 

or 

A*=A-ai-ir-^r -^r 

-^f'p -^f'p 

-^9" 9 -^9^9' 

- T^/V - A/V + A/V 

- ^9'P9 - ^9'P9' + i9'P (? + 9*) 

+ A/'Y + ^rW + W/"(4/- 11^ + 14 yy'-lin). 
Therefore, if we substitute in this equation 0^)8,7 for the series which they repre- 
sent, we shall have 

[11] A* = A-<rQa + ^fi + ^y + ^rf+iff'p(<l + q') 

+ iA»(3?'-2?^+3y'«) + T^/>«(4/-lly«+14y^-ll^«)). 

Art. 26, p. 41. Derivation of formula [12]. 

We form the expressions {q — ^y + f^ — r^^ — 2(g — q')r cob^ and (y — 5^) r sin ^. 
Then, since 

(y-y')« + r*-r'» = a* + c*- J*=2a(?cos^*, 

2 (y — /) r cos ^ = 2 ae cos B, 



72 NOTES 

we have 

(y — y')«+r«—r"— 2 (y — /)rcos^= 2 ac (cos J?* — cos 5). 

We have also 

{g — q') r sini^ = ae sin B. 

Subtracting (4) on page 68 from [1] on page 36, and adding this difference to 
{q — g'yy we obtain 

(1) (9 - y')" + r» - r", or 2 ae cos B* 

If we multiply [3] on page 37 by 2 (y — g')y we obtain 

(2) 2 (y — q') r cos ^, or 2 ae? cos jff 

= 2gi3-q') + iri^q{g-g')+fyqiq-g') +(1/"-^/°')/? (?-?') + etc. 

Subtracting (2) from (1), we have 

(3) 2 ac (cos 5* — cos 5) 

= -2/(y-/)»(i/' + i/> + (i/"- tV/'")/^+ etc. 

Multiplying [2] on page 36 by {q — q'), we obtain at once 

(4) (y — y')r8in^, or acsin.^ 

=P(3-9')Q-irf-i/'pf- (i/" + A/°')/?'+ etc. 

We now set B*—B = Cy whence B*=B + I^^ and therefore 

cos -B* = cos .B cos i — sin .B sin i. 

This becomes, after expanding cos £ and sin £ and neglecting powers of £ above the 

second, 

cos-B 
cos-B*— cos.B = 2 — •f— sin-B. i. 

Multiplying both members of this equation by 2 ac, we obtain 

(5) 2 a{?(cos-B* — cos -B) = — ac cos -B . f — 2 ac sin .B • i. 



NOTES 78 

Again, let li=R^+R^+R^+etc.^ where the ^s have the same meaning as before. 
Hence, replacing the terms in (5) by the proper series and neglecting terms aboye the 
sixth degree, we have 

(6) g{9-9')B,' + 2p(q-q'){l-irfH^.+R>+Ji:i 

= 2/ (q - g'Y i\r + U'P + (i/" - tV/°')/ 

+ (iA*'-iftr/'")(3?'+2??'+n). 
From this equation we find 

B,='p(q-q')afy+Wp{2q + q') + ih^{Sq'+2qq'+q'*) 

-iAr/'*(4/^+ 16?*+ 9??'+ 7n)- 
Therefore we have, correct to terms of the fifth degree, 

-i!V/'M4/+ 16j«+ 9y/+ 70), 
or, after factoring the last factor on the right, 

(7) B*=B-:^p{g-q')(i-ir(y+^+99'-^9")yi-ir-u'p-y''i^g+g') 

-U'Y-i9'p{^9 + 9')-ih<'{Sq' + 2qq'+q") 

+ TrHr/°'(-2y+22j»+8y^+4y'»)). 
The last factor on the right in (7) may be pat in the form : 

-i/'p -A/> 

+ ir Y + A/*Y' - tjW*"(2/+ 8 ?*+ 11 /'- 8 q^)y 

Finally, substituting in (7) cr, a, )8, y for the expressions which thej represent, we 
obtain, still correct to terms of the fifth degree, 

[12] ^*=^-<r(TVa+i/8 + ^y + TV/V 

- W/"(2/+ 8?«-8yy'+ ll/«)). 



74 NOTES 

Art. 26, p. 41. Derivation of formula [13]. 

Here we form the expressions (g — ^)' + r" — r* — 2 (y — j^) r' cos (ir — ^) and 
(j' — y') r' sin (w — 1^) and expand them into series. Since 

iq-q'f+r''-t*=a'+b*-e'=2ab COB €f*, 
2 (j — /)/ cos (w— 1^) = 2aJcos C, 
we have 

(y-/)« + r^'-r*- 2 (y-/)*^ cos (ir-i^) = 2 ab (cos C* - cos 0). 

We have also 

{q~g')r' sin (w — i^) = a J sin C. 

Subtracting (3) on page 68 from (4) on the same page, and adding the result to 
{S~g'y, we find 

(1) {q - g'y+ r"- r«, or 2 oi cos C* 

= -2q'(3-q-)- 1/>/ {f- r) - ^/Yif- q") - (§/"- ^r^^is"- 9") " etc. 

By priming the ^'s in formula [3] on page 37, we get a series for r* cos t^, or for 
— r'cos(ir— 1^). K we multiply this series for — r'cos(flr — i^) by 2{q — ^), we find 

(2) — 2 (y — /) r' cos (ir — 1^), or —2ab cos C 

= 2 is -^) (s' + irp'q' + i/y / + (I/" - :hr')pw + etc. 

And therefore, by adding (1) and (2), we obtain 

(3) 2 a* (cos C*— cos C) 

= - 2/ (y - q'Tiir + \fP + (i/"- ■hnf'r etc. 

+ iy°(? + 2y') + i/;,(y + 2^) 

+ (iA°-^/>«)(^+2yy'+3n). 
Bj priming the ;'s in [2] on page 36, we obtain a series for r'sinti/, or for 
r'sin(ir — tl/). Then, multiplying this series for r' sin (ir — tl/) by {q — ^)y we find 

(4) (j — j'') r' sin (ir — ^'), or abwisiC 

=P (?-?')(! -i/*^'- */>/*-(*/"+ A/^VV-etc. 

As before, let C*—C=l^, whence (7* = C^ + i, and therefore 

cos C?*= cos Ccos { — sin Csin {. 



NOTES 76 

Expanding cos { and sin £ and neglecting powers of C above the second, this equation 

becomes 

cos 
cos Cf* — cos C= 2 — • { — sin C. {, 

or, after multiplying both members by 2 a by 

(6) 2aJ(cost7* — cosd7)= — aJcos C. f — 2aJsint7. i. 

Again we put l^=E^+R^+Ii^+ etc., the Rq having the same meaning as before. 
Now, by substituting (2), (3), (4) in (5), and omitting terms above the sixth degree, 
we obtain 

from which we find 

^4=i'(?-?')(i/V + i/;'(? + 2?0+iA°(?" + 2y?'+3n 

-KV/°*(4/+7j*+9y^+16y'«)). 
Therefore we have, correct to terms of the fifth degree, 
(6) C*-0=piq-q')Qr + i/> + i/ V + iPX? + 2 /) 

+ iy°(? + 2?') + iA''(?*+2yy'+30 

- ^r'{^f+ 7?* + 9>^ + 16 y'«)). 

The last factor on the right in (6) may be written as the product of two factors, one 
of which is i(l—i/° (/+?*+ ??'+?'*)), and the other, 

2(i/'+i/> + i^°(? + 2y')+i/'y + i/;'(? + 2^) 

+ iA»(j«+3y'«+2y^)-Vff/''(-/+2y«+4y/+ll3/«)), 
or, in another form, 

-■hf'p -u'p 



76 NOTES 

Hence (6) becomes, on substituting <r, a,fi,y for the expressions which they represent, 
[13] ^* = C-<r(TVa + A/8+iy + TV/V 

Art. 26, p. 41. Derivation of formula [14]. 

This formula is derived at * once by adding formulae [11] , [12] , [13] . But, as 
Gauss suggests, it may also be derived from [6], p. 38. By priming the y's in [6] 
we obtain a series for {^ + j/). Subtracting this series from [6], and noting tiiat 
^ — ^'+i/^ + ir — t|r'=i4+-B + <7, we have, correct terms of the fifth degree, 

(1) A-\-B+C='n-p{s-q')(J--\-y'p-\-y'y-\-\g'p{3 + q') 

-\f*{f + 2 j« + 2 yj' + 2 q^y 
The second term on the right in (1) may be written 

+ */"(+ j' + yy' + n), 

of which the last factor may be thrown into the form : 

-if'p -%fp 
-y"f -\f"f +i/'y 

Hence, by substituting <r, a, ^, y for the expressions they represent, (1) becomes 
[14] .l+5+(7=ir + <r(ia+ij8+iy + i/'y 

+*/;>(? + ?') + (SA-'-ir') (?«-?/ + /')). 

Art. 27, p. 42. Omitting terms above the second degree, we have 

The expressions in the parentheses of the first set of formulsB for -4*, jB*, C* 
in Art. 27 may be arranged in the following manner: 

if -2q'+2qg'+ q") = i 2(y + q'')-(y + f) -{f-2qq' + q'^)), 
(p^ +f +2qq'-2q'^ = i-{f + q'') + 2{p' + q»)-{f-2qq' + q'»)y 



NOTES 77 

Now substituting a*, b\ (^ for {f—^qq'+ q'^)y (/^+ y'*), (p*+ f) respectively, and 
changing the signs of both members of the last two of these equations, we have 

-(j^ -2q'+2qq'+q'^) =(«« + ,.«- 2 ft«), 
-{f +q' +2y/-2y'«) = (a«+i«-2(?»). 

And replacing the expressions in the parentheses in the first set of formulsB for 
A*y B*y C* by their equivalents, we get the second set. 

Art. 27, p. 42. /° = — ^^ ' /" ~ ^> ^^^^j ^^7 ^® obtained directly, without the 

use of the general considerations of Arts. 25 and 26, in the following way. In the 
case of the sphere 

rf«^=cos«(;|-)-d>«+rfj*, 
hence 

n = cos(-|) =l-2^ + 24^-etc., 
I. e.j 

^^~2W ^''^245*' f'='9''=f'=9'=^^ [Wangerin.] 

Art. 27, p. 42, 1. 16. This theorem of Legendre is found in the M^moires (His- 
toire) de 1' Academic Royale de Paris, 1787, p. 358, and also in his Trigonometry, 
Appendix, § V. He states it as follows in his Trigonometry : 

The very slightly curved spherical triangle^ whose angles are Ay B, C and whose sides 
are a, d, c, always corresponds to a rectilinear triangle, whose sides a, i, c are of the same 
lengths y and whose opposite angles are A — ^e, B— ^e, C— ^e, e being the excess of the 
sum of the angles in the given spherical triangle over two right angles. 

Art. 28, p. 43, 1. 7. The sides of this triangle are Hohehagen-Brocken, Insel- 
berg-Hohehagen, Brocken-Inselberg, and their lengths are about 107, 85, 69 kilometers 
respectively, according to Wangerin. 

Art. 29, p. 43. Derivation of the relation between a- and a^. 

In Art. 28 we found the relation 

A*=A--^a{2a+fi + y). 
Therefore 

sin il*= sin -4 cos (r^o-{2 a+fi + y)) — cos A sin (-j^ a-{2a+fi+ y)), 

which, after expanding cos (-j^ a{2a+/3+ y)) and sin (-j^ (r{2a+fi + y)) and reject- 
ing powers of (tV^(2 «i+)8 + y)) above the first, becomes 



78 NOTES 

(1) sin A* = BmA-co8A. i'^<r{2a+/3 + y)% 

correct to terms of the fourth degree. 

But, since a and a-* differ only by terms above the second degree, we may replace 
in (1) a by the value of a-*, ^bcaixxA'^. We thus obtain, with equal exactness, 

(2) sinil = sinil*(l + 3^J(?co8^.(2a + i8 + y)). 

Substituting this value for sin -4 in [9], p. 40, we have, correct to terms of the sixth 
degree, the first formula for a given in Art. 29. Since ibccoaA*^ or b^ + c^—c^y 
differs from 2 be cos A only by terms above the second degree, we may replace 2 be cos A 
in this formula for a by b^ + c^ — a^. Also <r* = ^ be sin A*. Hence, if we make 
these substitutions in the first formula for a-, we obtain the second formula for <r 
with the same exactness. In the case of a sphere, where a=/8 = y, the second 
formula for a reduces to the third. 

When the surface is spherical, (2) becomes 

sinil = sinil* (1 + g be cos A). 
And replacing 2bccosA in this equation by {b^+ <^— a*), we have 

sin .1 = sin ^* (1 + ^ ( ft« + c* - a*)), 
or 

S^ = (l + E (»•+••-«■))• 

And likewise we can find 

fg^ = (l + 3^(««+o«-*')), ?^ = (l + ^(a.+ j.-^)). 

Multiplying together the last three equations and rejecting the terms containing a* 
and a', we have 

FinaUj, taking the square root of both members of this equation, we have, with the 
same exactness, 

1 _i_ " / » -L jj -i_ j\ / / siu^ .sin.g .sin<7 \ 
<^ = l + 24(^+^+^) = >H sin^*. sin ^*. sing* )- 

The method here used to derive the last formula from the next to the iMt 
formula of Art. 29 is taken from Wangerin. 



NEUE 



ALLGEMEINE UNTERSUCHUNGEN 



UBER 



DIE KRUMMEN FLACHEN 



C1825] 



PUBLISHED POSTHUMOUSLY IN GAUSS'S WORKS, VOL. VIII, 1901. PAGES 408-443 



NEW GENERAL INVESTIGATIONS 



OF 



CURVED SURFACES 



[1825] 



Although the real purpose of this work is the deduction of new theorems con- 
cerning its subject, nevertheless we shall first develop what is already known, partly 
for the sake of consistency and completeness, and partly because our method of treat- 
ment is different from that which has been used heretofore. We shall even begin by 
advancing certain properties concerning plane curves from the same principles. 

1. 

In order to compare in a convenient manner the different directions of straight 
lines in a plane with each other, we imagine a circle with unit radius described 
in the plane about an arbitrary centre. The position of the radius of this circle, 
drawn paraUel to a straight Une given in advance, represents then the position of that 
line. And the angle which two straight lines make with each other is measured by 
the angle between the two radii representing them, or by the arc included between 
their extremities. Of course, where precise definition is necessary, it is specified at 
the outset, for every straight line, in what sense it is regarded as drawn. Without 
such a distinction the direction of a straight line would always correspond to two 
opposite radii. 

2. 

In the auxiliary circle we take an arbitrary radius as the first, or its terminal 
point in the circumference as the origin, and determine the positive sense of measur- 
ing the arcs from this point (whether from left to right or the contrary) ; in the 
opposite direction the arcs are regarded then as negative. Thus every direction of a 
straight line is expressed in degrees, etc., or also by a number which expresses them 
in parts of the radius. 



82 KARL FRIEDRIC5H GAUSS 

Such lines as differ in direction by 360°, or by a multiple of 360®, have, there- 
fore, precisely the same direction, and may, generally speaking, be regarded as the 
same. However, in such cases where the manner of describing a variable angle is 
taken into consideration, it may be necessary to distinguish carefully angles differing 
by 360°. 

If, for example, we have decided to measure the arcs from left to right, and if 
to two straight lines /, V correspond the two directions i, L\ then i'— i is the angle 
between those two straight lines. And it is easily seen that, since L' — L falls 
between — 180° and + 180°, the positive or negative value indicates at once that V 
lies on the right or the left of /, as seen from the point of intersection. This will 
be determined generally by the sign of sin(i'— i). 

If a a' is a part of a curved line, and if to the tangents at a^a' correspond 
respectively the directions a, a', by which letters shall be denoted also the corres- 
ponding points on the auxiliary circles, and if A^ A' be their distances along the arc 
from the origin, then the magnitude of the arc a a' or A' — A is called the amplitude 
of a a'. 

The comparison of the amplitude of the arc a a' with its length gives us the 
notion of curvature. Let I be any point on the arc a a', and let X, A be the same 
with reference to it that a, A and a', A' are with reference to a and a'. If now 
aX or A — -4 be proportional to the part a/ of the arc, then we shall say that a a' is 
uniformly curved throughout its whole length, and we shall call 

A-.4 
al 

the measure of curvature, or simply the curvature. We easily see that this happens 
only when a a' is actually the arc of a circle, and that then, according to our defini- 
tion, its curvature will be ± -? if r denotes the radius. Since we always regard r 

as positive, the upper or the lower sign will hold according as the centre lies to the 
right or to the left of the arc a a' {a being regarded as the initial point, a' as the 
end point, and the directions on the auxiliary circle being measured from left to 
right). Changing one of these conditions changes the sign, changing two restores it 
again. 

On the contrary, if A — il be not proportional to «/, then we call the arc non- 
uniformly curved and the quotient 

iL-A 



NEW GENERAL INVESTIGATIONS OP CURVED SURFACES [1826] 88 

may then be called its mean curvature. Curvature^ on the contrary^ always presup- 
poses that the point is determined, and is defined as the mean curvature of an element 
at this point; it is therefore equal to 

rfA 
dat 

We see, therefore, that arc, amplitude, and curvature sustain a similar relation to each 
other as time, motion, and velocity, or as volume, mass, and density. The reciprocal 
of the curvature, namely, 

dal 

is called the radius of curvature at the point /. And, in keeping with the above 
conventions, the curve at this point is called concave toward the right and convex 
toward the left, if the value of the curvature or of the radius of curvature happens 
to be positive ; but, if it happens to be negative, the contrary is true. 

3. 

If we refer the position of a point in the plane to two perpendicular axes of 
coordinates to which correspond the directions and 90^, in such a manner that the 
first coordinate represents the distance of the point from the second axis, measured in 
the direction of the first axis ; whereas the second coordinate represents the distance 
from the first axis, measured in the direction of the second axis ; if, further, the inde- 
terminates x, y represent the coordinates of a point on the curved line, % the length 
of the line measured from an arbitrary origin to this point, ^ the direction of the 
tangent at this point, and r the radius of curvature ; then we shall have 

dx = cos ^ . d%^ 
dy = sin ^ . d%^ 
_ ds 

If the nature of the curved line is defined by the equation F = 0, where F is a 
function of x, y, and if we set 

d V=pdx + qdy, 
then on the curved line 

pdx + qdy=Q. 

Hence 

|>oos^ + y8in^=0, 



84 EABL FBEEDBIOH GATJ8S 

and therefore 

tan<i = — ^» 
^ 9 
We have also 

If, therefore, we set, according to a well known theorem, 

dp=Pdx+ Qdy, 

dq=Qdx + Rdtfj 
then we have 

(P cos" ^ + 2 C cos ^ sin ^ + 5 sin* ^ds = {pBmj> — q cos ^) rf^, 

therefore 

1 _ P cos" ^ + 2 g cos ^ sin ^ +^ sin' ^ 

r~ p&ia<f^ — qQOS<f^ ^ 

or, since 

cos<^=^^^^^^y Bin^ = ^^^^^j; 

^l_ Pg'-2gjpy+iZ/ 

4. 

The ambiguous sign in the last formula might at first seem out of place, but 
upon closer consideration it is found to be quite in order. In fact, since this expres- 
sion depends simply upon the partial differentials of F, and since the function V itself 
merely defines the nature of the curve without at the same time fixing the sense in 
which it is supposed to be described, the question, whether the curve is convex 
toward the right or left, must remain undetermined until the sense is determined by 
some other means. The case is similar in the determination of <f> by means of the 
tangent, to single values of which correspond two angles differing by 180°. The 
sense in which the curve is described can be specified in the following different ways. 

I. By means of the sign of the change in :r. If :r increases, then cos <f> must be 
positive. Hence the upper signs will hold if q has a negative value, and the lower 
signs if q has a positive value. When 3^ decreases, the contrary is true. 

II. By means of the sign of the change in ^. If y increases, the upper signs 
must be taken when p is positive, the lower when p is negative. The contrary is 
true when y decreases. 

III. By means of the sign of the value which the function V takes for points 
not on the curve. Let 8:r, 8y be the variations of z, y when we go out from the 



NEW GENERAL INVESTIGATIONS OF CURVED SURFACES [1826] 86 

curve toward the right, at right angles to the tangent, that is, in the direction 
^ + 90^ ; and let the length of this normal be 8/t>. Then, evidently, we have 

8ar = 8/>.cos(<^ + 90°), 
8y = 8/>.sin(^ + 90^), 
or 

8 a? = — 8 /> . sin ^, 

8y = + 8 /> . cos (f>. 

Since now, when 8/t> is infinitely small, 

hV=pha; + qBy 

= {—p 8m<f>+ qcoH<f>)Sp 
= ,^BpV(y + f) 

and since on the curve itself V vanishes, the upper signs will hold if F, on passing 
through the curve from left to right, changes from positive to negative, and the con- 
trary. If we combine this with what is said at the end of Art. 2, it follows that the 
curv^e is always convex toward that side on which V receives the same sign as 

Pf—iQpq+Ej^. 

For example, if the curve is a circle, and if we set 

then we have 

p = 2x, q = 2y, 

P = 2, G = 0, E = 2, 

Pq'-2Qpq+Bf=Si/'+Su^=S(^, 

r = ±: a 
and the curve will be convex toward that side for which 

as it should be. 

The side toward which the curve is convex, or, what is the same thing, the signs 
in the above formulsB, will remain unchanged by moving along the curve, so long as 

ST 
&p 

does not change its sign. Since F is a continuous function, such a change can take 
place only when this ratio passes through the value zero. But this necessarily pre- 
supposes that p and q become zero at the same time. At such a point the radius 



86 EABL FBIEDRICH GAUSS 

of curvature becomes infinite or the curvature vanishes. Then, generally speaking, 
since here 

— psin^ + ycos^ 

will change its sign, we have here a point of inflexion. 

5. 

The case where the nature of the curve is expressed by setting y equal to a 
given function of x^ namely, ^ = JT, is included in the foregoing, if we set 

V=X-y. 
If we put 



then we have 



therefore 



dX= X' dx, dX' = X" dx, 

P=X', G = 0, R = % 

1 X" 



~r"~(l+jr«)*' 

Since q is negative here, the upper sign holds for increasing values of x. We can 
therefore say, briefly, that for a positive X" the curve is concave toward the same 
side toward which the y-axis lies with reference to the a;-axis; while for a negative 
X" the curve is convex toward this side. 

6. 

K We regard x, y as functions of ^, these formulae become still more elegant. 
Let us set 

dx _^ dx^ _ 

57 "" ^' 57" "~ ^ ^ 

^ — / iML— if 

ds'^^' da ~y • 



Then we shall have 



or 



2:' = cos ^, y' "= siii ^> 

r " r 



NEW GENERAL INVESTIGATIONS OF CURVED SURFACES [1826] 87 



or also 
so that 

represents the curvature, and 



the radius of curvature. 



1 

7. 



We shall now proceed to the consideration of curved surfaces. In order to repre- 
sent the directions of straight lines in space considered in its three dimensions, we 
imagine a sphere of unit radius described about an arbitrary centre. Accordingly, a 
point on this sphere will represent the direction of all straight lines parallel to the 
radius whose extremity is at this point. As the positions of all points in space 
are determined by the perpendicular distances x^ y, z from three mutually perpendicu- 
lar planes, the directions of the three principal axes, which are normal to these 
principal planes, shall be represented on the auxiliary sphere by the three points 
(1), (2), (3). These points are, therefore, always 90° apart, and at once indicate the 
sense in which the coordinates are supposed to increase. We shall here state several 
well known theorems, of which constant use will be made. 

1) The angle between two intersecting straight lines is measured by the arc [of 
the great circle] between the points on the sphere which represent their directions. 

2) The orientation of every plane can be represented on the sphere by means 
of the great circle in which the sphere is cut by the plane through the centre parallel 
to the first plane. 

3) The angle between two planes is equal to the angle between the great cir- 
cles which represent their orientations, and is therefore also measured by the angle 
between the poles of the great circles. 

4) If a;, y, z ; a/, y , z' are the coordinates of two points, r the distance between 
them, and L the point on the sphere which represents the direction of the straight 
line drawn from the first point to the second, then 

of = z + r cos(l)i, 
y = y + r cos(2)i, 
2/ = + r cos(3)i. 

5) It follows immediately from this that we always have 

cos»(i)i; + co8*(2)i; + oos*(3)i; = i 



88 EABL FRIEDBICH GAIJBS 

[and] also, if L' is any other point on the sphere, 

cos(l)i . cos(l)i' + cos(2)i . cos(2)Z' + cos(3)i . cos(3)i' = cosii'. 

We shall add here another theorem, which has appeared nowhere else, as far as 
we know, and which can often be used with advantage. 

Let i, i', i", L'" be four points on the sphere, and A the angle which LV 
and i'i" make at their point of intersection. [Then we have] 

cos LL' . cos Z^i'" - cos ZZ" . cos Z'Z'" = sin Zi'" . sin Z'Z" • cos A. 

The proof is easily obtained in the following way. Let 

AL=t, AL' = f, AL" = 1f', AL'" = 1f''^ 

we have then 

cos LL' = cos t coaf + sin t aiaf cos -4, 
cos i"X'" = cos r cos f' + sin f sin f' cos A, 
cos iX" = cos i cos f^ + sin ^ sin f^ cos -4, 
cos L' L'" = cos f cos ^" + sin ^^ sin <"' cos A. 

Therefore 

cos ZZ' cos i"i'" - cos ii" cos LT' 

= coaA I cos < cos f sin <" sin f" + cos <" cos <"' sin < sin ^^ 

— cos t cos ^' sin ^ sin f^' — cos ^ cos f" sin < sin ^' } 
= cos A (cos ^ sin f — cos ^" sin t) (cos ^ sin ^' — cos <" sin f) 
= cos ^ sin (^" -< ) sin {r - f) 
= co&AHmLr''smL'r\ 

Since each of the two great circles goes out from A in two opposite directions, 
two supplementary angles are formed at this point. But it is seen from our analysis 
that those branches must be chosen, which go in the same sense from L toward L'" 
and from Z' toward L'\ 

Instead of the angle A, we can take also the distance of the pole of the great 
circle LL'^' from the pole of the great circle Z'Z". However, since every great circle 
has two poles, we see that we must join those about which the great circles run in 
the same sense from L toward i'" and from Z' toward Z", respectively. 

The development of the special case, where one or both of the arcs LL'^' and 
L' L'' are 90°, we leave to the reader. 

6) Another useful theorem is obtained from the following analysis. Let £, £', 
L'^ be three points upon the sphere and put 



NEW GENERAL INVESTIGATIONS OF CURVED SURFACES [1825J 89 

cos L (1) = a?, cos L (2) = y, cos L (3) = «?, 
cos V (1) = a/, cos L' (2) = y , cos Z' (3) = ^, 

cos i;"(l) = ^', cos Z" (2) = y", cosi"(3) =0^'. 

We assume that the points are so arranged that they run around the triangle 
included by them in the same sense as the points (1), (2), (3). Further, let X be 
that pole of the great circle L' V which lies on the same side as L. We then have, 
from the above lemma, 

y'zT'-z' y" = sin L' L" . cos X(l), 
0^ a/' - 2:' /' = sin Z' i" . cos X(2), 
7fy"-y'7f'=^mL' L" . cos X(3). 

Therefore, if we multiply these equations by a?, y, z respectively, and add the pro- 
ducts, we obtain 

ip/e" + ^/'0 + ^'y 0^ - a:y V -TfyJ'- 2fyz = sin Z'i" . cos X i, 

wherefore, we can write also, according to well known principles of spherical trigo- 
nometry, 

sini'i".sinii''.8ini' 

= sinZ'i''.siniX' .sini" 

= sin L' i" • sin Z' Z" . sin Z, 

if Z, L', L'' denote the three angles of the spherical triangle. At the same time we 
easily see that this value is one-sixth of the pyramid whose angular points are the 
centre of the sphere and the three points X, L\ L'' (and indeed posUivey if etc.). 

8. 

The nature of a curved surface is defined by an equation between the coordinates 
of its points, which we represent by 

Let the total differential of / (a?, y, z) be 

Pdx + Qdy+Edzy 

where P, Q, It are functions of a?, y, z. We shall always distinguish two sides of the 
surface, one of which we shall call the upper, and the other the lower. Generally 
speaking, on passing through the surface the value of / changes its sign, so that, as 
long as the continuity is not interrupted, the values are positive on one side and nega- 
tive on the other. 



90 KARL FRIEDRICH GAUSS 

The direction of the normal to the surface toward that side which we regard as 
the upper side is represented upon the auxiliary sphere by the point L. Let 

cos L{1) =Xy cos i(2) = F, cos Z(3) = Z. 

Also let ds denote an infinitely small line upon the surface ; and, as its direction is 
denoted by the point X on the sphere, let 

cos X(l) = (y cos X(2) = 17, cos X(3) = J, 

We then have 

dz = ( d9j dy = 7) dsy dz = I dsy 

therefore 

and, since XX must be equal to 90^, we have also 

Xi + Y7i+Zi=(i. 

Since P, Q, R, Xy F, Z depend only on the position of the surface on which we take 
the element, and since these equations hold for every direction of the element on the 
surface, it is easily seen that P, Qy R must be proportional to X, F, Z. Therefore 

P=X,jLy Q=r,jL, R = Z,i, 



Therefore, since 



and 



li=PX+QY+RZ 



or 

^=±l/(P«+C* + iP). 

If we go out from the surface, in the direction of the normal, a distance equal to 
the element 8/t>, then we shall have 

8a?=JSr8p, 8y = F8/>, hz = Zhp 

and 

8/=P8iP + Qhy + BSg=iiSp. 

We see, therefore, how the sign of /x depends on the change of sign of the value of 
/ in passing from the lower to the upper side. 

9. 

Let us cut the curved surface by a plane through the point to which our nota- 
tion refers; then we obtain a plane curve of which ds is an element, in connection 
with which we shall retain the above notation. We shall regard as the upper side of 
the plane that one on which the normal to the curved surface lies. Upon this plane 



NEW GENERAL INVESTIGATIONS OP CURVED SURFACES [1825] 91 

we erect a normal whose direction is expressed by the point S of the auxiliary 
sphere. By moving along the curved line^ X and L will therefore change their posi- 
tions, while S remains constant, and \L and XS are always equal to 90^. Therefore 
X describes the great circle one of whose poles is S. The element of this great circle 

will be equal to — , if r denotes the radius of curvature of the curve. And again, 

T 

if we denote the direction of this element upon the sphere by X', then X' will evi- 
dently lie in the same great circle and be 90° from X as well as from S. If we 
now set 

cos X'(l) = f ', cos X'(2) = V, cos X'(3) = i', 

then we shall have 

H=e'i, i,=VT' <'f=fT' 

since, in fact, |, i;, { are merely the coordinates of the point X referred to the centre 
of the sphere. 

Since by the solution of the equation /(a?, y, ^) = the coordinate z may be 
expressed in the form of a function of or, y^ we shall, for greater simplicity, assume 
that this has been done and that we have found 

2r=jP(a?,y). 

We can then write as the equation of the surface 

z—F{xyy) = % 
or 

From this follows, if we set 

dF{x,y) = tdx + u rfy, 

where t, u are merely functions of a? and y. We set also 

dt =Tdx + Udy, du = Udx + Vdy. 

Therefore upon the whole surface we have 

dz^=^tdx'\' udy 
and therefore, on the curve, 

Hence difiPerentiation gives, on substituting the above values for d(y drj, dl^ 

{V-t€'-uri')^=^dt + 7,du 

=^{i'T+2ir,U+7,^V)ds, 



92 KABL FBIEDBICH GAUSS 

or 

cos i X' 

10. 

Before we further transform the expression just found, we will make a few 
remarks about it. 

A normal to a curve in its plane corresponds to two directions upon the sphere, 
according as we draw it on the one or the other side of the curve. The one direc- 
tion, toward which the curve is concave^ is denoted by X', the other by the opposite 
point on the sphere. Both these points, like L and S, are 90^ from X, and there- 
fore lie in a great circle. And since S is also 90*^ from X, Si = 90° — ZX', or 
=iX'-90°. Therefore 

cos i X' = ± sin 8X, 

where sin Si is necessarily positive. Since r is regarded as positive in our analysis, 
the sign of cosZX' will be the same as that of 

And therefore a positive value of this last expression means that L\' \b less than 
90°, or that the curve is concave toward the side on which lies the projection of the 
normal to the surface upon the plane. A negative value, on the contrary, shows that 
the curve is convex toward this side. Therefore, in general, we may set also 

r sin 8j& ' 

if we regard the radius of curvature as positive in the first case, and negative in 
the second. SZ is here the angle which our cutting plane makes with the plane 
tangent to the curved surface, and we see that in the different cutting planes passed 
through the same point and the same tangent the radii of curvature are proportional 
to the sine of the inclination. Because of this simple relation, we shall limit our- 
selves hereafter to the case where this angle is a right angle, and where the cutting 



NEW GENERAL INVESTIGATIONS OP CURVED SURFACES [1826] 98 



plane, therefore, is passed through the normal of the curred surface, 
for the radius of curvature the simple formula 



Hence we have 



11. 

Since an infinite number of planes may be passed through this normal, it follows 
that there may be infinitely many different values of the radius of curvature. In this 
case Ty JJ^ F, Z are regarded as constant, ^, 'ij, ^ as variable. In order to make the 
latter depend upon a single variable, we take two fixed points Jf, JIT 90° apart on the 
great circle whose pole is L. Let their coordinates referred to the centre of the sphere 
be a, ^, y ; a', ^, 7/. We have then 

cos X(l) = cos \M . cos i!f(l) + cos \M! . cos -W(l) + cos \L . cos X(l). 
If we set 



then we have 



and the formula becomes 



and likewise 



Therefore, if we set 



we shall have 



If we pat 



cosXJ!/'= sin^, 

f = a cos ^ + a' sin ^, 

7j =^ cos ^ + jS' sin ^, 
i = y cos ^ + 7/ sin ^. 

^ = (aa'r+ (a'iSH- aff^U^- ^ffV)Z, 



- ==-4 cos'^ + 2 ^ cos ^ sin ^ + <7sin'^ 

A + C . A-C ^ . . ^ . « . 
= — 2 ' 2 — cos 2 ^ +5 sm 2 ^. 

A — C 



94 EARL FfilEDKIGH GAUSS 

^ (J 

where we may assume that E has the same sign as — s — ' then we have 

^ = |(il+<7)+^cos2(^-^). 

It is evident that ^ denotes the angle between the cutting plane and another plane 
through this normal and that tangent which corresponds to the direction M. Evidently, 

therefore, - takes its greatest (absolute) value, or r its smallest, when <l> = 0; and - 

its smallest absolute value, when ^ = ^ + 90°. Therefore the greatest and the least 
curvatures occur in two planes perpendicular to each other. Hence these extreme 

values for - are 
r 



i(A+0)±M{^)'+S'} 



Their sum is A + C and their product is AC — B*, or the product of the two extreme 

radii of curvature is 

1 

~AC-B'' 

This product, which is of great importance, merits a more rigorous development. 
In fact, from formulso above we find 

AC-B*={afi:-fiay{TV-U*)2P. 
But from the third formula in [Theorem] 6, Art. 7, we easily infer that 



therefore 



Besides, from Art. 8, 



AC-B* = Z*{TV-IP). 



Z=± 



therefore 



1 
1/(1+ <»+»«)' 



TV— IP 
AC-B*^ 



Just as to each point on the curved surface corresponds a particular point L on 
the auxiliary sphere, by means of the normal erected at this point and the radius of 



NEW GENERAL INVESTIGATIONS OP CURVED SURFACES [1825] 96 

the auxiliary sphere parallel to the normal, so the aggregate of the points on the 
auxiliary sphere, which correspond to all the points of a line on the curved surface, 
forms a line which will correspond to the line on the curved surface. And, likewise, 
to every finite figure on the curved surface will correspond a finite figure on the 
auxiliary sphere, the area of which upon the latter shall be regarded as the measure 
of the amplitude of the former. We shall either regard this area as a number, in 
which case the square of the radius of the auxiliary sphere is the unit, or else 
express it in degrees, etc., setting the area of the hemisphere equal to 360^. 

The comparison of the area upon the curved surface with the corresponding 
amplitude leads to the idea of what we call the measure of curvature of the sur- 
face. If the former is proportional to the latter, the curvature is called uniform; 
and the quotient, when we divide the amplitude by the surface, is called the measure 
of curvature. This is the case when the curved surface is a sphere, and the measure 
of curvature is then a fraction whose numerator is unity and whose denominator is 
the square of the radius. 

We shall regard the measure of curvature as positive, if the boundaries of the 
figures upon the curved surface and upon the auxiliary sphere run in the same sense ; 
as negative, if the boundaries enclose the figures in contrary senses. K they are not 
proportional, the surface is non-uniformily curved. And at each point there exists a 
particular measure of curvature, which is obtained from the comparison of correspond- 
ing infinitesimal parts upon the curved surface and the auxiliary sphere. Let da be 
a surface element on the former, and dX the corresponding element upon the auxiliary 
sphere, then 

dt 

dtr 

will be the measure of curvature at this point. 

In order to determine their boundaries, we first project both upon the 2?^-plane. 
The magnitudes of these projections are ZdtTj Zdl,. The sign of Z will show whether 
the boundaries run in the same sense or in contrary senses around the surfaces and 
their projections. We will suppose that the figure is a triangle ; the projection upon 
the 2?^-plane has the coordinates 

^>y ; ^ + rf^> y^ dy ; x + BXf y+ By. 

Hence its double area will be 

2 Zd(r — dx.Bff — dy. Bz. 

To the projection of the corresponding element upon the sphere will correspond the 
coordinates : 



96 EABL FBIEDBICH GAUSS 

X, F, 

dX dX dY dY 

X+-^-dx + -^-dy, Y^^-dx-^-^-dy, 
From this the double area of the element is found to be 

The measure of curvature is, therefore, 

dX dY dX dY 



dx dff dy dx 



= fti. 



Since 



we have 



therefore 



X = -tZ, Y=-uZ, 

dX=-Z^{l+ v?)dt +Z^tu . duy 
dY = + ZUu.dt-Z^{l + f)dUy 

?X dY 

tt. =z« (2' r- i7«) ((1 + ^) (1 + u«) - ^tt«) 

=Z\TV-lP)ll-\-f+i^) 
=Z\TV-IF) 
_ TV-IP 
~(l + <»+u«)" 

the very same expression which we have found at the end of the preceding article. 
Therefore we see that 



I 

1 



and 



NEW GENERAL INVESTIGATIONS OP CURVED SURFACES [1825] 97 

"The measure of curvature is always expressed by means of a fraction whose 
numerator is unity and whose denominator is the product of the maximum 
and minimum radii of curvature in the planes passing through the normal." 

12. 

We will now investigate the nature of shortest lines upon curved surfaces. The 
nature of a curved line in space is determined, in general, in such a way that the 
coordinates a:, y, z of each point are regarded as functions of a single variable, which 
we shall call w. The length of the curve, measured from an arbitrary origin to this 
point, is then equal to 



m%h{%h&-^«- 



K we allow the curve to change its position by an infinitely small variation, the varia- 
tion of the whole length will then be 

dw dw ^ dw dw dw ^ dw 





dx dy 

dw ^ dw 



^m^m-m ^^^h^^h^m 



•\-hz.d 



>1 { ^^h ( 




The expression under the integral sign must vanish in the case of a minimum, as we 
know. Since the curved line lies upon a given curved surface whose equation is 

Pdx-^Qdy^-Rdz^^, 

the equation between the variations hx^ 8y, 80 

P84; + C8y + i280 = O 

must also hold. From this, by means of well known principles, we easily conclude 
that the differentials 



98 KARL FRIEDRICH GAUSS 

m 

dx dy 

dw J dw 



d "^l!^ d. 



A{{^h{%hm) ^\^%h{%hm\ 



d. 



dz 
dw 



4{{^h{%hm) 



must be proportional to the quantities P, Qy R respectively. If ds is an element of 
the curve; X the point upon the auxiliary sphere, which represents the direction of 
this element; L the point giving the direction of the normal as above; and f, ly, J; 
X, Yj Z the coordinates of the points X, L referred to the centre of the auxiliary 
sphere, then we have 

dx = Sd8y dy=yid8y dz = l^d8y 

Therefore we see that the above differentials will be equal to rff, drj, rf{. And since 
P, Qy R are proportional to the quantities -X, F, Z^ the character of the shortest line 
is such that 

d^ dvi rf{ 

13. 

To every point of a curved line upon a curved surface there correspond two 
points on the sphere, according to our point of view; namely, the point X, which 
represents the direction of the linear element, and the point £, which represents the 
direction of the normal to the surface. The two are evidently 90° apart. In our 
former investigation (Art. 9), where [we] supposed the curved line to lie in a plane, 
we had two other points upon the sphere; namely, S, which represents the direction 
of the normal to the plane, and X', which represents the direction of the normal to 
the element of the curve in the plane. In this case, therefore, S was a fixed point 
and X, X' were always in a great circle whose pole was S. In generalizing these 
considerations, we shall retain the notation S, X', but we must define the meaning of 
these symbols from a more general point of view. When the curve 8 is described, 
the points Z, X also describe curved lines upon the auxiliary sphere, which, gener- 
ally speaking, are no longer great circles. Parallel to the element of the second line, 



NEW GENEEAL mVESTIGATIONS OP CURVED SURFACES [1825] 99 

we draw a radius of the auxiliary sphere to the point X', but instead of this point 
we take the point opposite when X' is more than 90° from L. In the first case, we 
regard the element at X as positive, and in the other as negative. Finally, let S be 
the point on the auxiliary sphere, which is 90° from both X and X', and which is so 
taken that X, X', S lie in the same order as (1), (2), (3). 

The coordinates of the four points of the auxiliary sphere, referred to its centre, 
are for 



L 


X Y 


Z 


X 


f 1? 


i 


X' 


f 1?' 


i' 


s 


a j8 


y- 



Hence each of these 4 points describes a line upon the auxiliary sphere, whose elements 
we shall express by dLj rfX, rfX', rfS* We have, therefore, 

di = i'd\. 



In an analogous way we now call 



d\ 



ds 
the measure of curvature of the curved line upon the curved surface, and its reciprocal 

ds 

the radius of curvature. If we denote the latter by p, then 

pdi=(^dSf 
pd7i = ri'dSf 
pdi = i'ds. 

If, therefore, our Une be a shortest line, f', i^', i' must be proportional to the 
quantities X, JT, Z. But, since at the same time 

we have 

e=±x, v=±r, (;'=±z, 

and since, further, 

f X+Vr+i'Z=co8X'Z 

= ±1, 



100 KASL FRIEDRICH GAUSS 

and since we always choose the point X' so that 

X'X<90% 
then for the shortest line 

X'X=0, 

or X' and L must coincide. Therefore 

pd7i = Y dsy 
pdl^ =ZdSy 

and we have here, instead of 4 curved lines upon the auxiliary sphere, only 3 to con- 
sider. Every element of the second line is therefore to be regarded as lying in the 
great circle XX. And the positive or negative value of p refers to the concavity 
or the convexity of the curve in the direction of the normal. 

14. 

We shall now investigate the spherical angle upon the auxiliary sphere, which 
the great circle going from L toward X makes with that one going from L toward 
one of the fixed points (1), (2), (3) ; e. ff.y toward (3). In order to have something 
definite here, we shall consider the sense from £(3) to £X the same as that in which 
(1), (2), and (3) lie. If we call this angle ^, then it follows from the theorem of Art. 
7 that 

sinjC(3) . siniX . sin^= Y^—Xr/f, 

or, since L\ = 90® and 

sin i(3)= ^/(JP* + F«) = V{\ - Z^y 
we have 



Furthermore, 



or 



and 



sin ^ - p^-yT+jiy • 
sinX(3) . sinZX . cos^ = {, 

tan ^ = J = -y • 



NEW GENERAL INVESTIGATIONS OF CIIRVED SUEPACES [1826] 101 
Hence we have 

^ {n-Xr,y+c [ 

The denominator of this expression is 

=-z*i»+(i--^(i-0 + i' 

or 

, , _ iYd^-iXdri + {X',i-Y^)di-riidX+^JidY 
»9 \—Z* 

We verify readily by expansion the identical equation 

'tii{x^ + r* +z») + rz(f » + 1,« + (;») 

= {X^-\-Yy,-\-Zi){Z7i + Yi)-^{Xi-Z^){X7i-Y^) 
and likewise 

^{(x* + J* +^*) +^z(^« + V + D 

We have, therefore, 

^i==-YZ+{Xi-Z^){X'^- n), 

U==-xz^ {Y^-X'n){Yi-Zyi). 

Substituting these values, we obtain 

dif> = Y^^ {YdX-XdY) + ^^M^Z^±1 

+ ^^Z^^ \dC-iXi-Z^)dX-{Yl-Zr,)dY\. 

Now 

XdX+YdY+ZdZ=0, 

(dX +7fdY + i^dZ =-Xdi-Ydri-ZdC. 
On substituting we obtain, instead of what stands in the parenthesis, 

dC-Z{Xd(+Ydri+ZdC). 
Hence 

d<i>^Y^{YdX-XdY) + j^ \i:Y-r,X'Z+ $XYZ\ 

-j^\iX +'nXYZ-iY^Z\ 
+ dCivX-(Y). 



102 



EARL FRIEDRICH GAUSS 



Since, further, 

nX*Z- iXrZ= iiX^Z+ ij T*Z+ { ZYZ 

= 'ilZ{\-Z*)-\-iYZ\ 
riX¥Z-(Y^Z=-(X*Z-i:XZ'-iT*Z 

=-€Z{i-z*)-cxz*, 

our whole expression becomes 

dif> = -Y^{rdX-Xdr) 

+ {i:Y-r,Z)di + iiZ-CX)dri + {rfX-ir)dC. 

15. 

The formula just found is true in general, whatever be the nature of the curve. 
But if this be a shortest line, then it is clear that the last three terms destroy each 
other, and consequently 

But we see at once that 

-^^{XdY-YdX) 

is nothing but the area of the part of the auxiliary sphere, which is formed between 
the element of the line Z, the two great circles drawn through its extremities and 




<z) 



PP' 




<3) 



(1) W 



(1) w 



(3) 




11) 



(3), and the element thus intercepted on the great circle through (1) and (2). This 
surface is considered positive, if L and (3) lie on the same side of (1) (2), and if the 



NEW GENERAL INVESTIGATIONS OP CURVED SUEPACES [1826] 108 



direction from P to P' is the same as that from (2) to (1) ; negative, if the contrary 
of one of these conditions hold ; positive again, if the contrary of both conditions be 
true. In other words, the surface is considered positive if we go around the circum- 
ference of the figure LL'P'P in the same sense as (1) (2) (3); negative, if we go 
in the contrary sense. 

K we consider now a finite part of the line from L io L' and denote by ^, j/ 
the values of the angles at the two extremities, then we have 

f = ^ + Area ii'P'P, 

the sign of the area being taken as explained. 

Now let us assume further that, from the origin upon the curved surface, infinitely 
many other shortest lines go out, and denote by A that indefinite angle which the 
first element, moving counter-clockwise, makes with the first element of the first line ; 
and through the other extremities of the diflFerent curved lines let a curved line be drawn, 
concerning which, first of all, we leave it undecided whether it be a shortest line or 
not. K we suppose also that those indefinite values, which 
for the first line were <^, <^', be denoted by t^, i// for each of 
these lines, then ^' — ^ is capable of being represented in 
the same manner on the auxiliary sphere by the space 
LL\P\P, Since evidently t^ = ^— -4, the space 

LL\P\P'L'L = t//-t/^-f+<^ 

= LL\L'L^'L'L\P\P\ 

If the bounding line is also a shortest line, and, when prolonged, makes with 
LVyLL\ the angles B^B^; if, further, x^Xx denote the same at the points L\L\, 
that ^ did at L in the line LL', then we have 

X,= X + Area L'L\ P\P\ 
but 

therefore 

B-B-\-A=LL\L'L. 

The angles of the triangle LL'L\ evidently are 

A, 180« —B, B^y 




106 KAEL FRIEDRICH GAUSS 

dx _dy dz 

^""37' ^~a7' t-^^- 

The extremities of all shortest lines of equal lengths s correspond to a curved 
line whose length we may call t. We can evidently consider ^ as a function of 8 and 
6, and if the direction of the element of t corresponds upon the sphere to the point X' 
whose coordinates are ^', tj', ^', we shall have 

Consequently 

If? -ri?i?-rttjg^ a* dd^ds dd^ ds dd 

This magnitude we shall denote by u, which itself, therefore, will be a function of 6 and s. 
We find, then, if we diiferentiate with respect to s, 

~ aa* 'dQ^ aa» "a^"^ a«* *a^ 

because 

and therefore its differential is equal to zero. 

But since all points [belonging] to one constant value of ^ lie on a shortest line, 
if we denote by L the zenith of the point to which «, Q correspond and by X^ Yy Z 
the coordinates of L, [from the last formulae of Art. 13], 

if j9 is the radius of curvature. We have, therefore, 

du dz dv dz dt 

But 

xf + rv + Zi' = cos ix' = 0, 

because, evidently, \' lies on the great circle whose pole is L. Therefore we have 

— = 



NEW GENERAL INVESTIGATIONS OF CURVED SURFACES [1825] 107 

or u independent of «, and therefore a function of d alone. But for s = 0, it is evi- 

dt 
dent that < = 0, ^ = 0, and therefore w = 0. Whence we conclude that, in general, 

«* = 0, or 

cos XX' = 0. 

From this follows the beautiful theorem : 

"If all lines drawn from a point on the curved surface are shortest lines of 
equal lengths, they meet the line which joins their extremities everywhere at 
right angles." 
We can show in a similar manner that, if upon the curved surface any curved 
line whatever is given, and if we suppose drawn from every point of this line toward 
the same side of it and at right angles to it only shortest lines of equal lengths, the 
extremities of which are joined by a line, this line will be cut at right angles by 
those lines in all its points. We need only let in the above development represent 
the length of the ffiven curved line from an arbitrary point, and then the above calcu- 
lations retain their validity, except that u = for 8 = is now contained in the 
hypothesis. 

18. 

The relations arising from these constructions deserve to be developed stiU more 

dt 
fully. We have, in the first place, if, for brevity, we write m for ^> 

^^) aJ""^' Yb-"^' dl~^' 

(2) a^=^^^ al9==^^^ a^=^i^ 

(3) r +v +r =1, 

(4) r'+v* + i"=i, 

(5) ^r+i?v+a'=o. 

Furthermore, 

(6) X^ +7^ +Z^ =1, 



and 



(7) jc^+r, +zc=o, 

(8) xr + rvH-^i'=o, 

[9] { Y = H'-i^\ 



108 EABL FBIEDBIGH GAT7S8 

[10] { r,'=iX-iZ, 

C = ir-'nX; 



[11] 



\ r, = Zi'-XC, 

[ i=Xr,'- r^\ 



Likewise, ^> ^> y are proportional to -X, F, 2^, and if we set 

^^-„Y ^'^-r.Y ^^-.7 

Ys-P^> g^-/>r, s^-pZ» 

where - denotes the radios of curvature of the line 8, then 

^ ds ds ds 

By differentiating (7) with respect to s, we obtain 

_ ax dY dZ 






^t9 g^/ J[y/ 

We can easily show that ^ — > -z-^> -^ — also are proportional to Jf, JT, -^. In fact, ' 

[from 10] the values of these quantities are also [equal to] 

ZZ dY dX dZ dY dX \ 

"iTf"^^' <^"97''^"97' ^'di~'^'d7' ' 

therefore 

^3f' y9V_ .(YdY XdX\ dZ 

= 0, 
and likewise the others. We set, therefore, 

whence 



^->({(B'-0"Mfr} 



NEW GENERAL INVESTIGATIONS OP CURVED SURFACES [1826] 109 

and also 

9f' ail' dC 

^ 08 08 d8 

Further [we obtain], from the result obtained by differentiating (8), 
But we can derive two other expressions for this. We have 

ds ~dff V d8 ~ dff ds "a^J 

therefore [because of (8)] 

d( dri dC 

[and therefore, from (7),] 

, ^dX. dY dZ 

After these preliminaries [using (2) and (4)] we shall now first put m in the form 
and differentiating with respect to «, we have* 

ds ~d0' da d$' ds d0' ds 

^* da.dd^^ ds.dd^^ ds.dd 

+ f a^ + '?a^+fc dd 
~f d0^^d0^^ dd' 



*It is better to differentiate m'. [In &ct from (2) and (4) 



therefore 



dm _dz 9*« , dv ^tf . dz ^g 



ds dd ddds d$ ddds dd dOds 



110 KAHL FRIEDRICH GAUSS 

If we differentiate again with respect to «, and notice that 

^i _dipX) 



d8d0 30 



etc., 



and that 
we have 



X^' + Yri' + Zi' = Q, 



/ax ,^Y dZ\ 



/ax, ,ar , ^,azv/ .ax ^ ar ^ az\ 

I 

-/^M^ — lZMW-i-^— — — — — \r4-^— — — — — W "^ 

\de ds ds ddf '^\de ds ds ddf ^\de a« a« a^' 

[But if the surface element 4 

mdadd 

belonging to the point rr, y, 2? be represented upon the auxiliary sphere of unit radius 
by means of parallel normals, then there corresponds to it an area whose magnitude is 

\^\d8 dd dd dsf^^yds dd dd dsf'^^\ds dd dd dsf^^^^' 

Consequently, the measure of curvature at the point under consideration is equal to 

1 d^ml 
m ds^ J 



J 

1 



NOTES 111 



NOTES. 

The parts enclosed in brackets are additions of the editor of the German edition 
or of the translators. 

" The foregoing fragment, Neue allgemeine Untersuchungen uher die krummen Flachen, 
differs from the Disqumtiones not only in the more limited scope of the matter, but 
also in the method of treatment and the arrangement of the theorems. There [paper 
of 1827] Gauss assumes that the rectangular coordinates x, y, ;8f of a point of the sur- 
face can be expressed as functions of any two independent variables p and q^ while 
here [paper of 1825] he chooses as new variables the geodesic coordinates s and 6. 
Here [paper of 1825] he begins by proving the theorem, that the sum of the three 
angles of a triangle, which is formed by shortest Unes upon an arbitrary curved surface, 
differs frorii 180® by the area of the triangle, which corresponds to it in the represen- 
tation by means of parallel normals upon the auxiliary sphere of unit radius. From 
this, by means of simple geometrical considerations, he derives the fundamental theo- 
rem, that "in the transformation of surfaces by bending, the measure of curvature at 
every point remains unchanged." But there [paper of 1827] he first shows, in Art. 
11, that the measure of curvature can be expressed simply by means of the three 
quantities E, jP, G, and their derivatives with respect to p and y, from which follows 
the theorem concerning the invariant property of the measure of curvature as a corol- 
lary ; and only much later, in Art. 20, quite independently of this, does he prove the 
theorem concerning the sum of the angles of a geodesic triangle." 

Remark by Stackel, Gauss's Works, vol. viii, p. 443. 

Art. 3, p. 84, 1. 9. cos*<^, etc., is used here where the German text has cos<^^ 
etc. 

Art. 3, p. 84, 1. 13. j»*, etc., is used here where the German text has /?jp, etc. 

Art. 7, p. 89, U. 13, 21. Since XZ is less than 90°, cosXZ is always positive 
and, therefore, the algebraic sign of the expression for the volume of this pyramid 
depends upon that of sin L'L^\ Hence it is positive, zero, or negative according as 
the arc Z'Z" is less than, equal to, or greater than 180°. 

Art. 7, p. 89, 11. 14-21. As is seen from the paper of 1827 (see page 6), Gauss 



112 NOTES 

corrected this statement. To be correct it should read : for which we can write also, 
according to well known principles of spherical trigonometry, 

sin LL' . sin Z' . sin L* L"'=- sin Z'Z" . sin L" . sin L" L=' sin L" L . sin Z . sin LL'y 

if Ly L'y L" denote the three angles of the spherical triangle, where L is the angle 
measured from the arc LL'' to LL', and so for the other angles. At the same time 
we easily see that this value is one-sixth of the pyramid whose angular points are 
the centre of the sphere and the three points X, L'^ L'' ; and this pyramid is positive 
when the points Z, L'y V are arranged in the same order about this triangle as the 
points (1), (2), (3) about the triangle (1) (2) (3). 

Art. 8, p. 90, 1. 7 fr. bot. In the German text V stands for / in this equation 
and in the next Une but one. 

Art. 11, p. 93, 1. 8 fr. bot. In the German text, in the expression for B^ {a^ + a^f) 
stands for {a'fi+ afi^). 

Art. 11, p. 94, 1. 17. The vertices of the triangle are Jlf, -8f, (3), whose coor- 
dinates are ct, ^, y; a', jS', 7/; 0, 0, 1, respectively. The pole of the arc MJif on 
the same side as (3) is X, whose coordinates are X, F, Z. Now applying the formula 
on page 89, line 10, 

a//' - /a/' = sin rZ" cos X(3), 

to this triangle, we obtain 

ajS' — iSa' = sin MM cos Z(3) 
or, since 

Jlfif = 90% and cos L{S)=±Z 
we have 

Art. 14, p. 100, 1. 19. Here JC, F, Z; f, 17, {; 0, 0, 1 take the place of a?, y,sf ; 
a/, y, 0^ ; a/', y", 0" of the top of page 89. Also (3), X take the place of L% L"^ and 
^ is the angle L in the note at the top of this page. 

Art. 14, p. 101, 1. 2 fr. bot. In the German text {IX—^XYZ-^^Y^ZX stands 
for {iX+iyXFZ-fr»J?}. 

Art. 16, p. 102, 1. 13 and the following. Transforming to polar coordinates, 
r, Qj 1^, by the substitutions (since on the auxiliary sphere r = 1) 

JT = sin ^ sin i/f, F= sin Q cosi/f, -2"= cos d, 
rfjr=sin^cos^rf^+ cos dsin^{]?0, rfjr= — sin^sin^rf^ + cos cos iff (/0, 

Z 

(1) = \—Z^ ^^^ ^ ^ YdX) becomes cos 6 d^. 



NOTES 118 

In the figures on page 102, PL and P'U are arcs of great circles intersecting in 
the point (3), and the element ZZ', which is not necessarily the arc of a great circle, 
corresponds to the element of the geodesic line on the curved surface. (2)PP'(1) 
also is the arc of a great circle. Here P^P = d^j Z= cos ^= Altitude of the zone 
of which LL* P'P is a part. The area of a zone varies as the altitude of the zone. 
Therefore, in the case under consideration, 

Area of zone _ Z 

Also 

Axe2,LL'P'P _d^ 

Area of zone ""2^* 

From these two equations, 

(2) Area L L'P'P = Z rfi/f, or cos Qd^. 

From (1) and (2) 

- 1^2 {XdY- YdX) = Area LL'P'P. 

Art. 15, p. 102. The point (3) in the figures on this page was added by the 
translators. 

Art. 15, p. 103, 11. 6-9. It has been shown that £?^ = Area LL^P'Py = dA^ say. 
Then 

<^' A 

fd<l>=fdA, 

ff> 

or 

<f}' — <f}=Ay the finite area LL' P'P. 

Art. 15, p. 103, 1. 10 and the following. Let Ay B'j B^ be the vertices of a 
geodesic triangle on the curved surface, and let the corresponding triangle on the 
auxiliary sphere be LL' L\Lj whose sides are not necessarily arcs of great circles. Let 
j4, B'y J?i denote also the angles of the geodesic triangle. Here B' is the supple- 
ment of the angle denoted by B on page 103. Let ^ be the angle on the sphere 
between the great circle arcs ZX,Z(3),i. ^., <^ = (3)ZX, X corresponding to the direc- 
tion of the element at A on the geodesic line AB'^ and let <^' = (3)Z'Xp X^ correspond- 
ing to the direction of the element at B' on the line AB'. Similarly, let ^= (3)Z^, 



114 



NOTES 




^ = (3)X\/Xi, ft, /ij denoting the directions of the elements at 
A, Bi, respectively, on the line A By And let x ~ (3) L' v, 
Xi=(3)Z'ii'j, V, v-y denoting the directions of the elements at 
B'f B^, respectively, on the line B' B.^. 

Then from the first formula on page 103, 

<^' - <^ = Area ZZ'P'P, 
I]/ — 1], = Area LL\P\P, 
Xi-x = AreaZ'Z',P',P', 

f - 1^ - (f - <^) - (Xi- x) = Area LL\P\P - Are&LL'P'P- Area L'L\P\P', 
or 
(1) {^-^) + (x-f ) + (f -xx) = AreaZX',!Z'Z. 

Since X, fi represent the directions of the linear elements at A on the geodesic 
lines A J?', A B^, respectively, the absolute value of the angle A on the surface is meas- 
ured by the arc /nX, or by the spherical angle [iLX. But c^ — i^ = (3)iiX — (3)Z/i 
= juXX. 
Therefore 



SimUarly 



Therefore, from (1), 



A = <!> — 1^. 

lSOo-B^ = -{x-n 



A+B' + B,-1S0'' = Area L L\ L' L. 

Art. 15, p. 103, 1. 19. In the aerman text ZZ'P'P stands for LL\P\F, 
which represents the angle t// — i/f. 

Art. 15, p. 104, 1. 12. This general theorem may be stated as follows : 

The sum of all the angles of a polygon of n sides, which are shortest lines 
upon the curved surface, is equal to the sum of (n — 2)lS0^ and the area of the 
polygon upon the auxiliary sphere whose boundary is formed by the points L which 
correspond to the points of the boundary of the given polygon, and in such a manner 
that the area of this polygon may be regarded positive or negative according as it is 
enclosed by its boundary in the same sense as the given figure or the contrary. 

Art. 16, p. 104, 1. 12 fr. bot. The zenith of a point on the surface is the cor- 
responding point on the auxiliary sphere. It is the spherical representation of the 
point. 

Art. 18, p. 110, 1. 10. The normal to the surface is here taken in the direction 
opposite to that given by [9] page 107. 



1 



I* 

I 



BIBLIOGRAPHY 



1 



BIBLIOGRAPHY. 

This bibliography ia limited to books, memoirs, etc., which use Gauss's method and which treat, more or less 
generally, one or more of the following subjects : curvilinear coordinates, geodesic and isometric lines, curvature of 
snrfitces, deformation of surfaces, orthogonal systems, and the general theory of surfiftces. Several papers which lie 
beyond these limitations have been added because of their importance or historic interest. For want of space, gener- 
ally, papers on minimal surfaces, congruences, and other subjects not mentioned above have been excluded. 

Generally, the numbers following the volume number give the pages on which the paper is found. 

0. B. will be used as an abbreviation for Comptes Bendus hebdomadaires des s^ces de PAoad^mie des 
Sciences. Paris. 



Adam, Paul. Sur les systdmes triples orthogonauz. Thesis. 
80 pp. Paris, 1887. 

Sur les surfaces isoihermiques il lignes de courbure 
planes dans un systdme ou dans les deux systdmes. 
Ann. de rlUiole Normale, ser. 8, vol. 10, 819-858, 1898 ; 
0. B., vol. 116, 1086-1089, 1898. 

Sur les sur&oes admettant pour lignes de courbure 
deux series de cercles g6odteiques orthogonaux. Bull, 
de la^Soc. Math, de France, vol. 22, 110-115, 1894. 

M^moire sur la deformation des surfistces. Bull, de la 
Soc Math, de France, vol. 28, 219-240, 1895. 

Sur la deformation des sur&ces. Bull, de la Soc. 
Math, de France, vol. 28, 106-111, 1895 ; 0. B., voL 
121, 551-558, 1895. 

Sur la deformation des sur&ces avec conservation des 
lignes de courbure. Bull, de la Soc. Math, de France, 
vol. 28, 195-196, 1895. 

Theordme sur la deformation des surfaces de transla- 
tion. BulL de la Soc. Math, de France, vol. 28, 204- 
209, 1895. 

Sur un probldme de deformation. BulL de la Soc. 
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Zur Theorie der inflnitesimalen Biegungsdeformationen 
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sohaften zu Mundien, vol. 27, 229-801, 1897. 



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126 



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CORRIGENDA ET ADDENDA. 

Art. 11, p. 20, 1. 6. The fourth E should be F. 

Art. 18, p. 27, 1. 7. For V {EG-F^) . dp . dO read 2 V{EG-F^) . dq . dO. 
The original and the Latin reprints lack the factor 2 ; the correction is made in all 
the translations. 

Art. 19, p. 28, 1. 10. For g read q. 

Art. 22, p. 34, 1. 5, left side; Art. 24, p. 36, 1. 5, third equation; Art. 24, 
p. 38, 1. 4. The original and Liouville's reprint have q for p. 

Note on Art. 23, p. 55, 1. 2 fr. hot. For p read q. 



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